Reinhold Remmert
Classical Topics in Complex Function Theory Translated by Leslie Kay
With 19 Illustrations
Springer
Reinhold Remmert Mathematisches Institut Westfiilische Einsteinstrasse 62 Mi.inster D-48149 Germany
Miinster
Leslie Kay (Translator) Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0123 USA
Editorial Board
S. Axler Department of Mathematics Michigan State University East Lansing. MI 48824 USA
F.W. Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA
P.R. Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA
Mathematics Subject Classification (1991): 30-01, 32-01 Library ()I Congress Cataloging-in-Publication Data Remmert, Reinhold I-unktionentheorie 2 Englishl Classical topics in complex function theory / Reinhold Remmert : translated by Leslie Kay. cm — (Graduate texts in mathematics : 172) Translation of Funktionentheorie il Includes bibliographical references and indexes ISBN 0-387-98221-3 (hardcoNer . alk. paper) 1. Functions of complex variables L Title. II Series. QA331.7 R4613 1997 515 - .9--dc2 97-10091
Printed on acid-free paper
1998 Springer-Verlag New York Inc All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc. 175 Fifth Avenue, New York, NY 10010, USA). except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar DI dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone. Production managed by Lesley Poliner: manufacturing supervised by Jeffrey Taub. Photocomposed pages prepared from the author's TEX files. Printed and bound by Edwards Brothers. Inc , Ann Arbor, MI.' Printed in the United States of America. 987654321 ISBN 0-387-98221-3 Springer-Verlag New York Berlin Heidelberg
SPIN 10536728
Max Koecher in
memory
Preface
Preface to the Second German Edition In addition to the correction of typographical errors, the text has been materially changed in three places. The derivation of Stirling's formula in Chapter 2, §4, now follows the method of Stieltjes in a more systematic way. The proof of Picards little theorem in Chapter 10, §2, is carried out following an idea of H. 1{61-Jig. Finally, in Chapter 11, §4. an inaccuracy has been corrected in the proof of Szegii's theorem. Oberwolfach, 3 October 1994
Reinhold Renim ert
Preface to the First German Edition Wer sich mit einer Wissenschaft bekannt machen will, darf nicht nur nach den reifen Priichten greifen — er muf3 sich da.rum bekiimmern. vie und wo sic gewachsen sind. (Whoever wants to get to know a science shouldn't just grab the ripe fruit he must also pay attention to how and where it grew.) .1. C. Poggendorf
Presentation of function theory with vigorous connections to historical development and related disciplines: This is also the leitmotif of this second volume. It is intended that the reader experience function theory personally
viii
Preface to the First German Edition
and participate in the work of the creative mathematician. Of course, the scaffolding used to build cathedrals cannot always be erected afterwards; but a textbook need not follow Gauss, who said that once a good building is completed its scaffolding should no longer be seen.' Sometimes even the framework of a smoothly plastered house should be exposed. The edifice of function theory was built by Abel, Cauchy, Jacobi, Riemann, and Weierstrass. Many others made important arid beautiful contributions; not only the work of the kings should be portrayed, but also the life of the nobles and the citizenry in the kingdoms. For this reason, the bibliographies became quite extensive. But this seems a small price to pay. "Man kann der studierenden Jugend keinen griii3eren Dienst erweisen ais wenn man sie zweckmd13ig anleitet, sich durch das Studium der Quellen mit den Fortschritten der Wissenschaft bekannt zu machen." (One can render young students no greater service than by suitably directing them to familiarize themselves with the advances of science through study of the sources.) (letter from Weierstrass to Casorati, 21 December 1868) Unlike the first volume, this one contains numerous glimpses of the function theory of several complex variables. It should be emphasized how independent this discipline has become of the classical function theory from which it sprang. In citing references, I endeavored -- as in the first volume — to give primarily original works. Once again I ask indulgence if this was not always successful. The search for the first appearance of a new idea that quickly becomes mathematical folklore is often difficult. The Xenion is well known: Allegire der Erste nur falsch, da schreiben ihm zwanzig Immer den Irrthum nach, ohne den Text zu besehn. 2 The selection of material is conservative. The Weierstrass product theorem, Mittag-Leffier's theorem, the Riemann mapping theorem, and Runge's approximation theory are central. In addition to these required topics, the reader will find – Eisenstein's proof of Euler's product formula for the sine; – Wielandt's uniqueness theorem for the gamma function; – an intensive discussion of Stirling's formula; - Iss'sa's theorem; 'Cf. W. Sartorius von Waltershausen: Gauf3 zum Cediichtnis, Hirzel, Leipzig 1856; reprinted by Martin Sdndig oHG, Wiesbaden 1965, p. 82. 2 Just let the first one come up with a wrong reference, twenty others will copy his error without ever consulting the text. [The translator is grateful to Mr. Ingo Seidler for his help in translating this couplet.]
Preface to the First German Edition
ix
- Besse's proof that all domains in C are domains of holomorphy;
Wedderburn's lemma and the ideal theory of rings of holomorphic functions: Estermann's proofs of the overconvergence theorem and Bloch's t he()rem; - a holomorphic imbedding of the unit disc in C'; - Gauss's expert opinion of November 1851 on Riemann's dissertation. An effort was made to keep the presentation concise. One worries, however:
Weiti uns der Leser auch fiir unsre Kiirze Dank'? Wohl kaum? Denn Kiirze ward durch Vielheit leider! lang. 3 Oberwolfach, 3 October 1994
3
Re tnhold Rensmc rt
1s the reader even grateful for our brevity? Hardly? For brevity, through
abundance, alas! turned long.
Preface to the First German Edition
Gratzas ago It is impossible here to thank by name all those who gave me valuable advice. I would like to mention Messrs. R. B. Burckel, J. Elstrodt, D. Gaier, W. Kaup, M. Koecher, K. Lamotke, K.-J. Ramspott, and P. Ullrich, who gave their critical opinions. I must also mention the Volkswagen Foundation, which supported the first work on this book through an academic stipend in the winter semester 1982-83. Thanks are also due to Mrs. S. Terveer and Mr. K. Schl6ter. They gave valuable help in the preparatory work and eliminated many flaws in the text. They both went through the last version critically and meticulously, proofread it, and compiled the indices.
.4dmee to the reader. Parts A, B, and C are to a large extent mutually independent. A reference 3.4.2 means Subsection 2 in Section 4 of Chapter 3. The chapter number is omitted within a chapter, and the section number within a section. Cross-references to the volume Funktzonentheorie I refer to the third edition 1992; the Roman numeral I begins the reference, e.g. 1.3.4.2. 4 No later use will be made of material in small print; chapters, sections and subsections marked by * can be skipped on a first reading. Historical comments are usually given after the actual mathematics. Bibliographies are arranged at the end of each chapter (occasionally at the end of each section); page numbers, when given, refer to the editions listed. Readers in search of the older literature may consult A. Gutzmer's German-language revision of G. Vivanti's Theorze der ezndeutigen Funktionen, Teubner 1906, in which 672 titles (through 1904) are collected.
'3 11n this translation, references, still indicated by the Roman numeral I, are to Theory of Complex Functions (Springer. 1991), the English translation by R. B. Burckel of the second German edition of Funktionentheorie I. Trans.]
Contents
Preface to the Second German Edition Preface to the First German Edition Acknowledgments Advice to the reader
vii Nrii
A
Infinite Products and Partial Fraction Series
1
1
Infinite Products of Holomorphic Functions §1 . Infinite Products 1. Infinite products of numbers 2. Infinite products of functions §2. Normal Convergence 1. Normal convergence Normally convergent products of holomorphic 2. functions 3. Logarithmic differentiation _ z 22 ) §3. The Sine Product sin7rz = rz 1. Standard proof 2. Characterization of the sine by the duplication formula Proof of Euler's formula using Lemma 2 3. 4*. Proof of the duplication formula for Euler's product , following Eisenstein 5. On the history of the sine product
3 4 6 7 7 9 12 12 14 15 16 17
xii
2
C'ont ents
18 19
Euler Partition Products 1. Partitions of natural numbers and Euler products 2. Pentagonal number theorem. Recursion formulas for p(n) and a(n) 3. Series expansion of Fri, ( 1 + qv z) in powers of z 4. On the history of partitions and the pentagonal number theorem §5*. Jacobi's Product Representation of the Series
20 22
.1(z, q) := (f2 zy 1. Jacobi 's theorem 2. Discussion of Jacobi's theorem 3. On the history of Jacobi's identity Bibliography
25 25 26 28 30
The Gamma Function §1. The Weierstra s s Function A(z) = zelz rt,,1(1 + z1v)e - '1 1. The auxiliary function
33 36
2. §2.
H(z) :=- z11 1 (1 + z/v)e - ziv The entire function A(z) := el' 11(z)
The Gamma Function 1. Properties of the F-function 2. Historical notes The logarithmic derivative 3. 4. The uniqueness problem 5. Multiplication formulas 6*. Holder's theorem 7*. The logarithm of the F-function §3. Elder's and Hankers Integral Representations of F(z) . . . 1. Convergence of Euler 's integral 2. Euler's theorem The equation 3*. 4*. Hankers loop integral §4. Stirling's Formula and Gudermann's Series 1. Stieltjes's definition of the function ii(z) 2. Stirling's formula 3. Growth of IF(s + iy)I for II -> pc 4-. Gudermann's series 5*. Stirling's series 6*. Delicate estimates for the remainder term 7*. Billet's integral Lindeliif's estimate 8*. §5. The Beta Function 1. Proof of Euler's identity 2. Classical proofs of Euler's identity Bibliography
24
36 37 39 39 41 42 43 45 46 47 49 49 51 52 53 55 56 58 59 60 61 63 64 66 67 68 69 70
Contents 3
Entire Functions with Prescribed Zeros §1. The Weierstrass Product Theorem for C 1. Divisors and principal divisors 2. Weierstrass products 3. Weierstrass factors 4. The Weierstrass product theorem Consequences 5. 6. On the history of the product theorem §2. Discussion of the Product Theorem Canonical products 1. 2. Three classical canonical products 3. The a-function 4. The p-function • 5* An observation of Hurwitz Bibliography
4* Holomorphic Functions with Prescribed Zeros §1. The Product Theorem for Arbitrary Regions 1. Convergence lemma 2. The product theorem for special divisors 3. The general product theorem 4. Second proof of the general product theorem 5. Consequences Applications and Examples 1. Divisibility in the ring 0(G). Greatest common divisors 2. Examples of Weierstrass products 3. On the history of the general product theorem 4. Glimpses of several variables §3. Bounded Functions on E and Their Divisors 1. Generalization of Schwarz's lemma 2. Necessity of the Blaschke condition 3. Blaschke products 4. Bounded functions on the right half-plane Appendix to Section 3: Jensen's Formula Bibliography 5 Iss'sa's Theorem. Domains of Holomorphy §1. Iss'sa's Theorem 1. Bers's theorem 2. Iss'sa's theorem 3. Proof of the lemma 4. Historical remarks on the theorems of Bers and Iss'sa Determination of all the valuations on M(G) . 5*.
xiii 73 74 74 75 76 77 78 79 80 81 82 83 85 85 86 89 89 90 91 92 92 93 94 94 96 97 97 99 99 100 100 102 102 104 107 107 108 109 109 110 111
xiv
Contents Domains of Holomorphy 1. A construction of Goursat 2. Well-distributed boundary sets. First proof of the existence theorem Discussion of the concept of domains of holomorphy 3. 4. Peripheral sets. Second proof of the existence
t heorem 5.
115 116 118
On the history of the concept of domains of
holomorphy 6.
Glimpse of several variables §3. Simple Examples of Domains of Holornorphy 1. Examples for E 2. Lifting theorem 3. Cassini regions and domains of holomorphy Bibliography
6
112 113
Functions with Prescribed Principal Parts §1. Mitt ag-Leffier's Theorem for C 1. Principal part distributions Mittag-Leffier series 2. 3. NIittag-Leffier's theorem 4. Consequences Canonical Mittag-Leffler series. Examples 5. 6. On the history of Mittag-Leffler's theorem for C . . §2. Mittag-Leffier's Theorem for Arbitrary Regions Special principal part distributions 1. Mittag-Leffler's general theorem 2. 3. Consequences 4. On the history of Mittag-Leffler's general theorem 5. Glimpses of several variables §3*. Ideal Theory in Rings of Holomorphic Functions 1. Ideals in 0(G) that are not finitely generated . . 2. Wedderburn's lemma (representation of 1) 3. Linear representation of the gcd. Principal ideal theorem 4. Nonvanishing ideals Main theorem of the ideal 'theory of 0(G) 5. On the history of the ideal theory of holomorphic 6. functions 7. Glimpses of several variables Bibliography
119 120 120 120 122 122 123 125 126 126 127 128 128 129 130 131 131 132 133 134 135 135 136 136 1 :3388 139 140 141 142
Contents
B 7
Mapping Theory The Theorems of Montel and Vii ah Monters Theorem 1. 'Montel's theorem for sequences 2. Proof of Writers theorem Montel's convergence criterion 3. 4. Vitali's theorem Pointwise convergent sequences of holontorphic 5. functions §2. Normal Families Montel's theorem for normal families 1. Discussion of Montel's theorem 2. 3. On the history of Montel's theorem Square-integrable functions and normal families 4*. §3*. Vitali's Theorem 1. Convergence lemma Vitali's theorem (final version) 2. On the history of Vitali's theorem 3. §4*. Applications of Vitali's theorem 1. Interchanging integration and differentiation Compact convergence of the I'-integral 2. 3. Miintz's theorem §5. Consequences of a Theorem of Hurwitz Bibliography §1.
8
The Riemann Mapping Theorem §1.
§2.
Integral Theorems for Homotopie Paths 1. Fixed-endpoint homotopic paths Freely homotopie closed paths 2. 3. Null hornotopy and null homology Simply connected domains 4. 5*. Reduction of the integral theorem 1 to a lemma . . Proof of Lemma 5* 6*. The Riemann Mapping Theorem 1. Reduction to Q-domains 2. Existence of holomorphic injections
3. 4. 5. 6. §3.
On 1. 2. 3.
Existence of expansions Existence proof by means of an extremal principle On the uniqueness of the mapping function Equivalence theorem the History of the Riemann Mapping Theorem Riernann's dissertation Early history From Carathéodory-Koebe to Fej6r-Riesz
xv
145 147 148 148 150 150 I 50 151 152 152 153 154 154 156 156 157 1 58 159 159 160 161 162 164 167 168 168 169 170 171 172 174 175 175 177 177 178 179 180 181 181 183 184
xvi
Content s
Carathéodory's final proof 1list orical remarks on uniqueness and boundary
4. 5.
behavior 6. Glimpses of several variables !sot ropy Groups of Simply Connected Domains 1. Examples 2. TIR' group A ut „G for simply connected domains
G C Nlapping radius. Nlonotonicity theorem
3*.
Appendix to Chapter 8: Carathéodory-Koebe Theory Simple Properties of Expansions 1. Expansion lemma 2. Admissible expansions. The square root method 3. The crescent, expansion W. The Carat héodory-Koebe Algorithm 1. Properties of expansion sequences
2. 3. 1. 5.
Convergence tlieorem Koebe families and Koebe sequences Summary. Quality of convergence Historical remarks. The competition between
Carathéodory and Koebe The Koebe Families /C,„ and /C-_,„ 1.
A lemma 2. The families C , „, and 1C. Bibliography for Chapter 8 and the Appendix
9 Automorphisms and Finite Inner Nlaps Inner Maps and Automorphisms 1. Convergent sequences in Hol G and Ant G
184 186 187 188 188 189 189 191 191 191 192 193 194 194 195 196 197 197 198 198 199 201 203 203 204
2.
r9
Convergence theorem for sequences of aut °morph isms 3. Bounded homogeneous domains 4*. limer maps of H and homotheties It prat ion of Inner Maps 1. Elementary properties 2. 11. Cart an's t heorem 3. The group Aut„G for Winded domains 4. The closed subgroups of the circle group Automorphisms of domains with holes. Annulus theorem Finite Holomorphic Maps 1. Three general properties 2. Finite inner maps of IE 3. Boundary lemma for annuli
204 205 206 206 207 207 208 209
.
!i3.
210 211 212 212 213
Contents
Finite inner maps of annuli Determination of all the finite maps between annuli §4*. Rad6's Theorem. Mapping Degree Closed maps. Equivalence theorem 1. Winding maps 2. Rad6's theorem 3. 4. Mapping degree 4. 5.
5.
Glimpses
Bibliography
xvii
215 216 217 217 218 9 19 220
221 221
C Selecta
223
10 The Theorems of Bloch. Picard, and Schottky §1. Bloch's Theorem 1. Preparation for t he proof 2. Proof of Bloch's t heorem Improvement of the bound by the solution of an 3*. extremal problem 4*. Ahlfbrs's theorem Landau's universal constant 5*. §2. Picard's Little Theorem
225 226 226 227 9 28
230 232 933 233 234 235 236 237
Representation of functions that omit two values 1. 2. Proof of Picard's little theorem Two amusing applications 3. §3. Schottky's Theorem and Consequences Proof of Schottky's theorem 1. Landau's sharpened form of 2. Picards little theorem 238 Sharpened forms of NIonters 3. and Vitali's theorems 239 240 §4. Picard's Great Theorem 1. Proof of Picard's great theorem 240 . . 240 On the history of the theorems of t his chapter 2. 241 Bibliography
11 Boundary Behavior of Power Series §1. Convergence on the Boundary 1. Theorems of Fatou, M. Riesz. and Ostrowski A lemma. of NI. Riesz 2. Proof of the theorems in 1 3. A criterion for nonextendibility 4. Bibliography for Section 1 §2. Theory of Overconvergence. Gap Theorem
243 244
244 245 247 248 249 249
xviii
Contents I.
2. 3. 4.
Overconvergent power series Ostrowski's overconvergence theorem Hadamard's gap theorem Porter's construction of overconvergent series
o.
On the history of the gap theorem 6. On the history of overconvergence 7. Glimpses Bibliography for Section 2 §3. A Theorem of Fatou-Hurwitz-P61ya I. IIurwitz's proof 2. Glimpses Bibliography for Section 3 §4. An Extension Theorem of Szeg6 1. Preliminaries for the proof of (Sz) 2. A lemma Proof of (Sz) 3. 4. An application 5. Glimpses Bibliography for Section 4
12 Runge Theory for Compact Sets §1. Techniques
249 250 252 253 254 255 255 256 257 258 259 259 260 260 262 263 263 265 266 267 268 269 271 272 273 273 275 276 278
1. Cauchy integral formula for compact sets 2. Approximation by rational functions 3. Pole-shifting theorem P. Runge Theory for Compact Sets 1. Runge's approximation theorem 2. Consequences of Runge's little theorem :3. Main theorem of Runge theory for compact sets §3. Applications of Runge's Little Theorem 1. Pointwise convergent sequences of polynomials that 278 do not converge compactly everywhere . 281 Holomorphic imbedding of the unit disc in C3 2. §4. Discussion of the Cauchy Integral Formula for 283 Compact Sets 284 1. Final form of Theorem 1.1 285 Circuit theorem 2. 287 Bibliography
13 Runge Theory for Regions §1. Runge's Theorem for Regions 1. 2. 3.
Filling in compact sets. Runge's proof of Mittag-Leffier's theorem Runge's approximation theorem Main theorem of Cauchy function theory
289 290 291
292 292
Contents
xix
293 On the theory of holes 4. 294 5. On the history of Runge theory 295 §2. Runge Pairs Topological characterization of Runge pairs 295 1. 296 2. Runge hulls 3. Homological characterization of Runge hulls. 297 The Behnke-Stein theorem 9 98 Runge regions 4. . 299 5*. Approximation and holomorphic extendibility 300 §3. Holomorphically Convex Hulls and Runge Pairs 300 1. Properties of the hull operator Characterization of Runge pairs by means of 2. holomorphically convex hulls Appendix: On the Components of Locally Compact Spaces. .31ira303 Bura's Theorem 303 1. Components Existence of open compact sets 304 2. 305 3. Filling in holes 4. Proof of Sura-Bura's theorem :3i (0)(ir) Bibliography 14 Invariance of the Number of Holes §1. Homology Theory. Separation Lemma 1. Homology groups. The Bet ti number Induced homomorphisms. Natural propert ies 2. 3. Separation of holes by closed paths §2. Invariance of the Number of Holes. Product Theorem for Units 1 On the structure of the homology groups 2. The number of holes and the Betti number 3. Normal forms of multiply connected domains (report) 4. On the structure of the multiplicative group 0(G) Glimpses 5. Bibliography Short Biographies Symbol Index Name Index Subject Index
309 309 310 311 312 313 313 314 315 316 318 318 321 329 331 337
Part A Infinite Products and Partial Fraction Series
1 Infinite Products of Holomorphic . Functions
Allgemeine Sdtze fiber die Convergenz der (mendlichen Producte sind zum grosser' Theile bekannt. (General theorems on the convergence of infinite products are for the most part well known.) — Weierstrass, 1851
Infinite products first appeared in 1579 in the work of F. Viet a (Opera, p. 400, Leyden, 1646); he gave the formula
for 7 (cf. [ZI, p. 104 and p. 118). In 1655 J. Wallis discovered the famous proditct 7r
2
22 4 • 4 66 2n 2n 1 • 3 3.5 5 • 7 • • • (271 — 1) • (2n — 1)
.
.
,
which appears in "Arithmetica infinitorum," Opera 1, p. 468 (cf. [Z]. p. 104 and p. 119). But L. Euler was the first to work systematically with infinite products and to formulate important product expansions; cf. Chapter 9 of his Introductio. The first convergence criterion is due to Cauchy, Cours d'analyse, p. 562 ff. Infinite products had found their permanent place in analysis by 1854 at t he latest. through Weierstrass ([Wei], p. 172 ff.).' I In 1847 Eisenstein. in his long-forgotten work [DJ, had already systematically used infinite products. He also uses conditionally convergent products (and
4
1. Infinite Products of Holomorphic Functions
One goal of this chapter is the derivation and discussion of Euler's product oo
Z
2
(1 _ ,)
sin 7rz = 7rz
v=1 for the sine function; we give two proofs in Section 3. Since infinite products are only rarely treated in lectures and textbooks on infinitesimal calculus, we begin by collecting, in Section 1, some basic facts about infinite products of numbers and of holomorphic functions. Normally convergent infinite products of functions are investigated in Section 2; in particular, the important theorem on logarithmic differentiation of products is proved.
ri
§1. Infinite Products We first consider infinite products of sequences of complex numbers. In the second section, the essentials of the theory of compactly convergent products of functions are stated. A detailed discussion of infinite products can be found in [Kn]. 1. Infinite products of numbers. If (a,) ,>k is a sequence of complex numbers, the sequence (FIi nv=k av)rt>k of partial products is called a(n) (infinite) product with the factors a,. We write a, or au or simply a„; in general, k = 0 or k = 1. fi If we now — by analogy with series — were to call a product TT a, convergent whenever the sequence of partial products had a limit a, undesirable pathologies would result: for one thing, a product would be convergent with value 0 if just one factor a, were zero; for another, a, could be zero even if not a single factor were zero (e.g. if la, < q < 1 for all v). We will therefore take precautions against zero factors and convergence to zero. We introduce the partial products
n k
ri
Pm ,n : = am a ni
an
=H
k <m < n,
1/=771
n
and call the product a, convergent if there exists an index in such that the sequence (pm,n)n>in has a limit an, O. series) and carefully discusses the problems, then barely recognized, of conditional and absolute convergence; but he does not deal with questions of compact convergence. Thus logarithms of infinite products are taken without hesitation, and infinite series are casually differentiated term by term; this carelessness may perhaps explain why Weierstrass nowhere cites Eisenstein's work.
§1. Infinite Products We then call a := akak + i duce the suggestive notation
5
am _lîzn, the value of the product and intro-
IT a, := akak+i
arn _ lam = a.
The number a is independent of the index m,: since a„, 0, we have a,, 0 0 for all n > Tn.; hence for each fixed I > in the sequence (pi,„)„>i also has a limit -a, o 0, and a = akak +i ai_i a,. Nonconvergent products are called divergent. The following result is immediate: A product T1 a, is convergent if and only if at most finitely many factors are zero and the sequence of partial products consisting of the nonzero O. elements has a limit The restrictions we have found take into account as well as possible the special role of zero. Just as for finite products, the following holds (by definition):
A convergent product
n a,
is zero if and only if at least one factor is
zero. We note further: If H 0, 1, =0 a v converges, then over, urn ar, 1 and lim a71 = 1.
:=
„ a, exists for
all n E N. More-
an
O. Then = a/po , Proof We may assume that a := fl a„ Since limpo, n _ i = a, it follows that lima = 1. The equality lima?, = 1 holds because, for all n, at, o 0 and an = anran.+1. Examples. a) Let ao := 0, a, := 1 for v > 1. Then fl at, = O. > 2. Then p2, n = (1 ± ); hence fl w>2 a, = b) Let a, := 1> 2. Then = ;17 ; hence Jim p2, n = O. The product c) Let a, := a„ is divergent (since no factor vanishes) although lim a,„ = 1.
In 4.3.2 we will need the following generalization of c): d) Let ao, a l , a2, ... be a sequence of real numbers with a n > 0 and E(1 - av ) = +oo. Then lim rE=0 a„ = O. Proof. 0 <po „ = non a„ < exp[- E on. (1 - a„)], n E N. since t < et -1 for all t E R. Since a„) = +co, it follows that limpo,„ = O.
E(1 -
It is not appropriate to introduce, by analogy with series, the concept
of absolute convergence. if we were to call a product fl a„ absolutely convergent whenever H la, I converged, then convergence would always imply absolute convergence — but 11(-1) 1' would be absolutely convergent without being convergent! The first comprehensive treatment. of the convergence theory of infinite products was given in 1889 by A. Pringsheim [PI.
6
1. Infinite Products of Holomorphic Functions
Exercises. Show:
a)n
11 3
+ 1
3
, 1,
v+ (-1) v=i
oc 2 b) n cos — 7r = - (Vieta's product). 2v 7r v=2
2. Infinite products of functions. Let X denote a locally compact metric space. It is well known that the concepts of compact convergence and locally uniform convergence coincide for such spaces; cf. 1.3.1.3. For a sequence L G C(X) of continuous functions on X with values in C, the (infinite) product f„ is called compactly convergent in X if, for every compact set K in X, there is an index in = m(K) such that the sequence pni,n frnfm+1 • • 71 > m, converges uniformly on K to a nonvanishing function fm. Then, for each point x E X,
H
:= fifif(x) E C
f
exists (in the sense of Subsection 1): we call the function f : X -4 C the limit of the product and write
f=
fl f1
;
then, on K. fiK = (folK) • • • • Cfm-11 10
fai
•
The next. two statements follow immediately from the continuity theorem 1.3. I .2. a) If 1-1 f„ converges compactly to f in X, then f is continuous in X and the sequence f„ converges compactly in X to 1. b) If f„ and g,, converge compactly in X. then so does fv g„:
H
n
H
fvg, = (H f (11g,) We are primarily interested in the case where X is a dornain 2 in C and all the functions fi, are holomorphic. The following is clear by the Weierstrass convergence theorem (cf. 1.8.4.1).
H
e) Let C be a domain in C. Every product f,, of functions f„ holomorphic in G that converges compactly in C has a limit f that is holomorphic G. (_, 2z \ -1 Examples. a) The functions f,, := (14_ 2,2z 1)1 > 1, are holomorphic k
2v+1
in the unit disc E. We have
2 p2 , „ = (1 + — z) (1+
3
-1 2z )
2n± 1
E 0(E);
hence
limp2,„ = 1 +
2
2 [As defined in Funktionentheone I. a region ("Bereich" in German) is a nonempty open subset of C. a domain ("Gebiet" in German) is a connected re-
gion. In consulting Theory of Complex Functions, the reader should be aware that there "Bereich" was translated as "domain" and "Gebiet" as "region." Trans.]
§2. Normal Convergence
7
fi, therefore converges compactly in E to 1 + 2z.
and the product
b) Let f„(z) E z for all y > O. The product ficc o f„ does not converge (even pointwise) in the unit disc E, since the sequence pm ,„ = z" -7" ." converges to zero for every rn.. -
We note an important sufficient Convergence criterion. Let , f„ E C(X), v > O. Suppose there exists an ni E N such that every function f,,, u > rn, has a logarithm log f,, E C(X). log f,, converges compactly in X to s E C(X), then 11 f,, converges If compa-ctly in X to fo ul exp s.
E >m
Ei „
log fi, converges compactly to s, the Proof. Since the sequence s„ := n, f„ = exps y, converges compactly in X to exp s. As sequence pr„,„ =f exp s does not vanish, the assertion follows. 3
r=n,
Normal Convergence
§ 2.
The convergence criterion 1.2 is hardly suitable for applications, since series consisting of logarithms are generally hard to handle. Moreover, we need a criterion — by analogy with infinite series — that ensures the compact convergence of all partial products and all rearrangements. Here again, as for series, "normal convergence" proves superior to "compact convergence." We recall this concept of convergence for series, again assuming the space f,,, f,, E C(X), is normally convergent in X to be locally compact: then X if and only if IfvIK < oo for every compact set K C X (cf. 1.3.3.2). Normally convergent series are compactly convergent; normal convergence is preserved under passage to partial sums and arbitrary rearrangements (cf. 1.3.3.1). The factors of a product ri f„ are often written in the form f„ = 1 + g„; by 1.2 a), the sequence g„ converges compactly to zero if F1 f„ converges compactly.
E
E
fi
1. Normal convergence. A product f,, with f,, = 1 + g,, E C( X) is g,, converges normally in called normally convergent in X if the series X. It is easy to see that
fiv,0 fi,
E
converges normally in X, then
- for every bzjection r : N mally in X;
N, the product 1-1, >0 fr()) converges 710T-
The simple proof that the compact convergence of s„ to s implies the compact convergence of exp s„ to exp s can be found in 1.5.9.3 (composition lemma). 3
8
1. Infinite Products of Holornorphic Functions - every subproduct ni>0 f converges normally in X; - the product converges compactly in X.
We will see that the concept of normal convergence is a good one. At the moment, however, it is not clear that a normally convergent product even has a limit. We immediately prove this and more: Rearrangement theorem. Let 111>0 f,, be normally convergent in X. N Then there is a function f : X -4 C su-ch that for every bzjection r :N the rearrangementriv>0 fr(, ) of the product converges compactly to f in X. Proof For w G E we have log (1 + w) = Ev>i
wv. It follows that
llog(1+w)1 < 1w1( 1 +1w1+1/0 1 2 + - '); hence log(1+w)1 m. For all such n, log f =
E (-1? -1 gr.", E C(K).
where 'log fu l K 5_ 21 gn1 •
We see that E,,,, log fi,1 K < Ev>m 19,1K < oo. Hence, by the rearrangement theorem for series (cf. I.C.4.3), for every bijection a of N„, In E N : n > ml the series Ev>in log f,(,) converges uniformly in K to log f,,. By 1.2, it follows that for such a the products [L >rn f,( ,) f,, converge uniformly in K to the same limit function. But an arbitrary bijection 7 of N (= permutation of N) differs only by finitely many transpositions (which have no effect on convergence) from a permutation :N with al(N7 ) = N„,. Hence there exists a function f : X C such that every product fro,) converges compactly in X to f.
nv>0
converge normally in X. Then the following
Corollary. Let f = riv>o statements hold. 1) Every product
fn
n
,, > ,,
L,
converges normally in X, and
f = fofi - • • fr,--
•
2) If N = Ur N„ is a (finite or infinite) partition of N into pairwise disjoint subsets AT 1 , . , N,: then every product FLEN,,, f converges normally in X and
f
= 11 11 f-) = K
1
E
§2. Normal Convergence
9
Products can converge compactly without being normally convergent, as is Ir - Vv. It is always true that shown, for example, by F1 v >1( 1 9v), 9v (1 + g2,_1)(1 + g2„) = 1; hence pi, = 1 for even n and pi, = 1 + for odd n. The product n u>i (1 + gy) thus converges compactly in C to 1. In this example the subproduct FL, (1. + g2'-1) is not convergent!
All later applications (sin é product, Jacobi's triple product, Weierstrass's factorial, general Weierstrass products) will involve normally convergent products. Exercises. 1) Prove that if the products fi f,, and F1 f,, converge normally in X, then the product 11(f,L) also converges normally in X. 2) Show that the following products converge normally in the unit disc E, and prove the identities
n ( 1 z2- )
1 1 z,
H[(1±z-)(1_?-1)]=1.
convergent products of holomorphic functions. The 0 holomorphic in G is locally finite in zero set Z(f) of any function f countably infinite (see 1.8.1.3). Z(f) is at most G; 4 hence For finitely many functions fo, ff, • • • E 0(G), f„ 0 0,
2. Normally
Z(fofi • - • fn) =UZ(fv) 0
and
oc(foff
fr,)
=
oc(f,,), c E G, 0
where o(f) denotes the order of the zero of f at c (1.8.1.4). For infinite products, we have the following result.
11
f„, f 0 0, be a normally convergent product in Proposition. Let f = G of functions holomorphic in G. Then f 0, Z(f) =UZ(f„), o c (f) =
E oc (f„)
for all C E G.
Proof. Let c E G be fixed. Since 1(c) = 1-1 f„(c.) converges, there ex0 for all y > n. By Corollary 1,1), ists an index n such that Mc) f E 0(G) by the Weierstrass f = fofi • • • fn_lj;„ where fT, := convergence theorem. It follows that n-
oc(f) = 4 Let
o(f,,)
with
oc (i;,) = 0
(since rf,-,(c) 0 0).
o
G be an open subset of C. A subset of G is locally finite in G if it intersects
every compact set in G in only a finite number of points. Equivalently, a subset of G is locally finite in G if it is discrete and closed in G.
10
1. Infinite Products of Holomorphic Functions
This proves the addition rule for infinite products. In particular, Z(f) = UZ(M. Since each f,, 0, all the sets Z(f i,) and hence also their countable union are countable; it follows that f O. Remark. The proposition is true even if the convergence of the product in G is only compact. The proof remains valid word for word, since it is easy to see that for every n the tail end 1;, - = f„ converges compactly in G.
n„„
We will need the following result in the next section. If f = flfi,, fi, E 0(G), is normally convergent in G, then the sequence = f,, E 0(G) converges compactly in G to 1.
11,n
Proof. Let f,„ $ O. Then A := Z(j) is locally finite in G. All the partial products E 0(G), n > m, are nonvanishing in G \ A and
1 pm,,i(z)
for all z E G
A.
Now the sequence 1/p7„,„_ 1 converges compactly in G \ A to 1/f„,. Hence, by the sharpened version of the Weierstrass convergence theorem (see 1.8.5.4), this sequence also converges compactly in G to 1.
Fr'
Exercise. Show that f = cos(z/2v) converges normally in C. Determine Z(f). Show that for each k E N \ {0} there exists a zero of order k of f and that
00
Li cos
( 2v —
—
2v
V=1
1
sin
2v — 1)
3. Logarithmic differentiation. The logarithmic derivative of a meromorphic function h e .A4(G), h 0, is by definition the function h' I h E M(G) (see also 1.9.3.1, where the case of nonvanishing holomorphic functions is discussed). For finite products h = h1h2...h7 , hp E M(G), we have t he h' 4. 2 h' ± m Addition formula: h1 h2 This formula carries over to infinite products of holomorphic functions. Differentiation theorem. Let f = fl fz, be a product of holomorphic functions that converges normally in G. Then f,' /f,, is a series of meromorphic functions that converges normally in G, and
E
§2. Normal Convergence
11
Proof 1) For all n E N (by Corollary 1,1)), -1
f
f=
fofi...fri-ifn,
E _f4.-
f with L., := H f„; hence —
fit
Since the sequence fr, converges compactly in G to 1 (cf. 2), the derivatives , converge compactly in G to 0 by Weierstrass. For every disc B with B C G there is thus an in E N such that all f„, n > m, are nonvanishing in B and the sequence j--Vi.„ E 0(B), n > m, converges compactly in B , / f,, converges compactly in G to l' if. to zero. This shows that 2) We now show that. E t f, converges normally in G. Let, g„ := f,, – 1. We must assign an index m to every compact set K in G so that every pole set P(t1 f„), y > m, is disjoint from K and
R
Ef
F, I/
(*) > rn
f
f
I/
m, are empty and minz K If (z)1 > for all u > m (this is possible, since the sequence f„ converges compactly to 1). Now. by the Cauchy estimates for derivatives, there exist a compact, set L D K in G and a constant M > 0 such that ig,1 K < MI.gvIL for all u (cf. I. 8 . 3 . 1 ). Thus ig./.fv1K 5 • (minzEK 1.fgz)l) < 211/1g,IL for r■ > m. Since EIg,f IL < oc by hypothesis, (*) follows.
The differentiation theorem is an important tool for concrete computations; for example, we use it in the next subsection to derive Euler's product for the sine, and we give another application in 2.2.3. The theorem holds verbatim if the word "normal" is replaced by "compact." (Prove this.) The differentiation theorem can be used to prove:
If f is h,olomorphic at the origin., then f can be represented uniquely in a disc B about 0 as a product
f(z) = bz h fl (i
which converges normally in
B
b„e), kb,
E C.
k
E
N,
to f
This theorem was proved in 1929 by J. F. Ritt [R]. It is not claimed that the product converges in the largest disc about. 0 in which f is holomorphic. There seem to be no compelling applications of this product expansion, which is a multiplicative analogue of the Taylor series.
12
1. Infinite Products of Holomorphic Functions
§3.
The Sine
Product
sin 7rz = 7rz FM:1 (1 —
The product fr,=1 (1-z2/u2) is normally convergent in C, since converges normally in C. In 1734 Euler discovered that
2 z H _ _v2
Z2 //1 2 )
En, z2/v2
oc
(1
sin rz =
)
Z
e C.
v=1
We give two proofs of this formula. 1. Standard proof (using logarithmic differentiation and the partial fraction decomposition for the cotangent). Setting f, := 1— z2 /v2 and f (z) := 7TZ 117_ 1 f„ gives
fLI fu =
2z
1 E 2z f' (z)I f (z) = + 2
and thus
2'
v=1
Z
— V2 •
Here the right-hand side is the function ir cot rz (cf. 1.11.2.1). As this is also the logarithmic derivative of sin rz, we have5 f (z) = c sin 7rz with c E C x . Since lim2 ..0 f (z) = 1 = 0 f it follows that c = 1. Substituting special values for z in (1) yields interesting (and uninteresting) formulas. Setting z := .1- gives the product formula 7r
,>0
2 2 4 4 6 6
--= 1
2v
2v
. . •" =11 3. 2v - 1 2v + 1 •• -J .-6- •g• 7
(Wallis, 1655).
= rivc- 2( 1 _ 7'7 ) (cf. Example 1.1, For z := 1, one obtains the trivial equality b); on the other hand, setting z := i and using the identity sin in = - e') give the bizarre formula
0. n (1+ _ 1 = v2
1
e7T
27re- 7
.
v=1
Using the identity sin z cos z =
sin 2z and Corollary 2.1, one obtains
oc,
cos rz sin 7rz =
7rz
11 1-
v.1
2z
2
.v
N>
= ITZ n
rl (1 (2:: i) 2 ), .=, (1- (-27z))2)
Let j u, g the same logarithmic derivative. that f/g E M(G) and (f/g )' E. O.
j -
L.y
Dmain G which have X . To prove this, note C
, (1 -
§3. The Sine Product sinwz = 7rz
z 2 /v 2 )
13
and hence Euler's product representation for the cosine:
In 1734-35, with his sine product, Euler could in principle compute all the numv -2 ', n 1,2, ... (cf. also 1.11.3.2). Thus it follows imbers ((2n) := mediately, for example, that ((2) = Since f(z) := FITL I (1 - z 2 /i2 ) = 22 1 _ ( EL I v-2)z2 tends compactly to f(z) := (sin rz)/(rz) = 1 - ' t: + • • , 0 it follows that fn"(0) = converges to ;21-f"(0) = ir 2 . Wallis's formula permits an elementary calculation of the Gaussian error integral fo'c e - '2 ids. For x"e - x2 dx, we have
217L = (n - 1)/7,2, n > 2
(integration by parts!).
Since /1 = -1, an induction argument gives
2 k 12k = 1 • 3 • 5 • ... • (2k - 1)/0,
(0)
k E N.
212k+1 = k!,
xn -1 (x + t)2 e - s 2 da- for all t E R, it follows that
Since /,,4.1 + 2tI„ +t2 1,1-1 =
/r2, < In _1 /n4-1;
hence
2/,2, < ne, i.
With (o) we now obtain
(k!) 2 2 7.2 4k + 2 2k + 1 2k-1-1 1
(k!) 2
r
J2k
s--
1
2k—ii2k-f =
4k
This can also be written 2 /2k =
(k W 4k + 2
(1 +
Ek),
with 0< Ek
1 < 7;7 YI£ •
Using (o) to substitute 10 into this yields
2/(1 =
[2 • 4 6 (2k) [1 3 • 5 • ... • (2k - 1)] 2 (2k + 1) (1 4- "). ] 2
From limEok = 0 and Wallis's formula, it follows that 241 = 2-7(- and hence that
0
fe2 dx=W.
This derivation was given by T.-J. Stieltjes: Note sur [intégrale r e - "' du, Nouv. Ann. Math. 9, 3rd ser., 479-480 (1890); Œuvres completes 2, 2nd ed., Springer, 1993. 263-264. Exercises. Prove: 1) lim
246
33.7
2) tior = 3)
2n . (2n+1) V
/Tr . 2 V '
FE-L,
ebz =
(a
—b)zei ( " +b)' 11-v -= (1 I (1 +
4, 2 ir 2z2
);
14
1. Infinite Products of Holomorphic Functions 4) cos(7z) — sin(-1vrz) =flt (I
+ ( 2771i riz).
2. Characterization of the sine by the duplication formula. We characterize the sine function by properties that are easy to verify for the product z 1-1(1 — z 2 /v2 ). The equality sin 2z = 2 sin z cos z is a Duplication formula: sin 27rz = 2 sin 7TZ sin 7r(z +
z E C.
In order to use it in characterizing the sine, we first prove a lemma. Lemma (Herglotz, multiplicative form). 6 Let G C C be a domain that contains an interval [0, r), r> 1. Suppose that y E 0(G) has no zeros in [0, r) and satisfies a multiplicative duplication formula
(*) g(2z) = cg(z)g(z + P
when
z, z +
2z E [0, r)
(with c E C x ).
Then g(z) = aebz with 1 = acelb . Proof. The function h := g'I g E M(G) is holomorphic throughout [0, r), and 2h(2z) = 2g1 (2z) g (2 z) = h(z) + h(z + ) whenever z, z+ 2z E [0, r). By Herglotz's lemma (additive form), h is constant. 6 It follows that g' = bg with b E C; hence g(z) = aebz. By (*), ace l b = 1. The next theorem now follows quickly. Theorem. Let f be an odd entire function that vanishes in [0, 1] only at 0 and 1, and vanishes to first order there. Suppose that it satisfies the Duplication formula: f (2z) = cf (z) f (z + ), z E C, Then 1(z) = 2c'
where c E C x .
7TZ.
Proof. The function g(z) := f (z) I sin 7rz is holomorphic and nowhere zero in a domain G D [0, r), r > 1; we have g(2z) = lcg(z)g(z+P. By Herglotz, f (z) = aebz sin 7rz with acelb = 2. Since f (— z) = f (z), it also follows that b = O.
6
We recall the following lemma, discussed in 1.11.2.2:
Herglotz's lemma (additive form). Let [0,r) C G with r > 1. Let h E 0(G) and assume that the additive duplication formula 2h(2z) = h(z)+ h(z + 1) holds when z, z + 2z E [0, r). Then h is constant. Proof. Let t E (1,r) and M := max.(11/,'(z)1 : z E [0, t)}. Since 4h'(2z) = h'(z) + /t / (z + D and lz and 1(z + 1) always lie in [O, t ] whenever z does, it follows that 4M < 2M, and hence that M = 0. By the identity theorem, h' = 0; thus h = const.
§3. The Sine Product sin Trz =
n - ( 1 _z2 / v2 )
15
We also use the duplication formula for the sine to derive an integral that
will be needed in the appendix to 4.3 for the proof of Jensen's formula: (1)
log sin rtdt = - log 2.
•i()
Proof. Assuming for the môment that the integral exists, we have i I 2 log sin 27rtdt = -1- log 2 + i log sin 7rtdi + i log sin 7r(( + (0)
f
..,* )
di.
•0
•0
0
Setting 7 := 2t on the left-hand side and 7- := t + 711 in the integral on the extreme right immediately yields (1). The second integral on the right in (0) exists whenever the first one does (set t + = 1 - T). The first integral [O. ;!2-j.' exists since g(t) := t-1 sin in is continuous and nonvanishing 3. Proof of Euler's formula using Lemma 2. The function
s(z)
00 z H
z2/1,2
v= 1 is entire and odd and has zeros precisely at the points of Z, and these s(z)/z = 1, Theorem 2 implies are first-order zeros. Since s'(0) = formula. This can be s satisfies a duplication 7r5(z) whenever that sin 7rz = verified immediately. Since s converges normally, it follows from Corollary 2.1 that rx
s(2z) = 2z
(+)
-
(2
9
- (2v) 2 ) H( 00
2s(z)II (1
f v=i
4Z 2
(2v — 1) 2 )
1
4z 2
(2v - 1) 2 ) •
A computation (!) gives (1- — 47,1 2 ) (1
4z 2 (2v - 1) 2 )
(2z + 1) 2 )
1 + 2z/(2v - l) ( 1+ 2z/(2v + 1)
v
i v2
> 1.
If we take Example a) of 1.2 into account, this yields 00
n
4z2
41v 2
u=i 1
(2v — 1) 2 )
= (1 + 2z)
II I
(I
( 2z
1 )2)
4 v+2
= 2s(z + Let f (t) = " g(t), t E N, where g is continuous and nonvanishing in [0, ri, r > O. Then LT log f (t) dt exists. This is clear since f; log t (It exists (a- log x — is an antiderivative, and lim6\0 log 6= 0). 7
16
1. Infinite Products of Holomorphic Functions
Thus (+) is a duplication formula: s(2z) = 4a- I s(z)s(z + 1), where a :=
[I (1 — t/4v2)
o.
0
This multiplicative proof dates back to the American mathematician E. H. Moore; a number of computations are carried out in his 1894 paper [M]. The reader should note the close relationship with Schottky's proof of the equation 00 / ( 1 1 it cot 7 z = - + E z .=_00 z + , ,1 ) in 1.11.2.1; Moore probably did not know Schottky's 1892 paper. 4*. Proof of the duplication formula for Euler's product, following Eisenstein. Long before Moore, Eisenstein had proved the duplication formula for s(z) in passing. In 1847 ([Ei], p. 461 ff.), he considered the apparently complicated product 00
i 1-e (i. +
E(w,z) := TT
z
n
IT
z ) . (i. + ) him
(1 +
z
) )
of two variables (w, z) E (C\Z) X C; here n e = lim n „, ri,n denotes the Eisenstein multiplication (by analogy with the Eisenstein summation Ee , which we introduced in 1.11.2). Moreover, if indicates that the factor with index 0 is omitted. The Eisenstein product E(w, z) is normally convergent in the (w, z)space (C\Z) x C, since n
n
V
(1
+W
Z
2
+
2w z )
ii 2 .... w 2
and EM I 1/(w 2 — v2 ) converges normally in C\Z (cf. 1.11.1.3). The function E(z, w) is therefore continuous in (C\Z) x C and, for fixed w, holomorphic in each z E C. Computations can be aimed out elegantly with E (z, w), and the following is immediate. Duplication formula. E(2w, 2z) = E(w, z)E (w + 1 , z).
Proof E(2w, 2z) = =
fi ]-4rie e
2z
)
r:. n (1
V. +
2v + 2W ) "e
2z 2v + 1 + 2w)
E(w,z)E(w +1,z).
0
Eisenstein used the (trivial, but astonishing!) formula
(*)
1+
z = (1 w ++ -z ) / (1 + v v+w v
(Eisenstein's trick)
to reduce his "double product" to Euler's product: E(w, z) =
+ z) s (w)
s(w
where
s(z) = z
H (1 v> 1
1( 1
§3. The Sine Product sin irz =
_ z2
17
Proof.
E(w, z) w
+z w
Fr,
w+z )
n—, oc "
00 / = (W Z)
lim rr
(
11
n
"
)
00 /
4_
,;2z)2
,
(w
w2 - -7)) v
s(w + z) s(w) The duplication formula for s(z) is now contained in the equation s(2w + 2z)
s(2w)
= E(2w,2z) = E(w, z)E(w + 2 , z) =
s(w + z) s(w + + z)
s(w)
s(w +
s(2w)= 2. it follows that Since s is continuous and lim w-o 8(w) s(w + + z) s(2w) = 2s(1) -1 8(z)s(z + ;1). s(w + z) s(2z) = lim w-O s(w) s(w +
The elegance of Eisenstein's reasoning is made possible by the second variable w. Eisenstein also notes (loc. cit.) that E is periodic in w: E(w + 1, z) = E(w, z) (proved by substituting v+1 for v); he uses E and s to prove the quadratic reciprocity law; the duplication formula appears there at the bottom of p. 462. Eisenstein calls the identity E(w, z) = s(in + z) I s(w) the fundamental formula and writes it as follows (p. 402; the interpretation is left to the reader):
z
1
am + 13
) =
sin
- z )/a , :in(3rf3
a, /3 E C, /3/a
5. On the history of the sine product. Euler discovered the cosine and sine products in 1734-35 and published them in the famous paper "De Summis Serierutn Reciprocarum" ([Eu], I-14, pp. 73-86); the formula 1
S4
36
1• 2•3•4-5
1•2•3•4•5•6•7
S2
2•3 = (1
s2) p2
1
s2 4p2
(1
s2 9p2
1
+ 52
16p2 • • •
(with p := 7r) appears on p. 84. As justification Euler asserts that the zeros of the series are p, p, 2p, 2p, 3p, -3p, etc., and that the series is therefore (by analogy with polynomials) divisible by 1- 1 + , 1 -1 + -21• etc 2p 2p "• In a letter to Euler dated 2 April 1737, Joh. Bernoulli emphasizes that this reasoning would be legitimate only if one knew that the function sin z -
-
1. Infinite Products of Holomorphic Functions
18
had no zeros in C other than nr, n E Z: "demonstrandurn esset null= contineri radicem impossibilem" ([C], vol. 2, p.16); D. and N. Bernoulli made further criticisms; cf. [Weil], pp. 264-265. These objections, acknowledged to some extent by Euler, were among the factors giving incentive to his discovery of the formula e" = cos z + i sin z; from this Euler, in 1743, derived his product formula, which then gives him all the zeros of cos z and sin z as a byproduct. Euler argues as follows: since lim(1 + z/n)' = ez and sin z =
1 2i
sin z = — lim pr,
(a),
where
-
p(w) := (1 + w)" - (1 - w) /1 .
For every even index n = 2m, it follows that pn (w) =
(*)
± tu n-2 ).
2nw(1 + w
The roots w of p„ are given by (1 +w) = ((1 - w), where ( = exp(2v/rt/n) is any nth root of unity; hence p2,. as an odd polynomial of degree n - 1, has the n - 1 distinct zeros 0, ±wi, ,±w„,_i, where
=
exp(2vri/n) - 1
exp(2viri/n) + 1
. V7r =7. tan — , v = 1, . , in - 1.
The factorization —1 P2rn. (W) =
2nw
rn —
W
(1.
2nw 1-1 (1 + w 2 cot 2 vr v=1
-
— 71
LI. I
then follows from (*). Thus n— I
sin z = z iim
n--•oc
— Z
2 ( 1 —
cot
(1
Since lim„_,c (.;17 cot 2` n1-r ) = j-yr , interchanging the limits yields the product formula. This last step can, of course, be rigorously justified (cf., for example, [VI, p. 42 and p. 56). An even simpler derivation of the sine product, based on the same fundamental idea, is given in [Nu], 5.4.3.
§4*.
Euler Partition Products
Euler intensively studied the product
Q(z, q) := ll( i ±qyz) = (1 + qz)(1 + q2 z)(1 + q3 z) as well as the sine product. Q(z , q) converges normally in C for every q E E since E Mt' < oc; the product is therefore an entire function in z, which
§4*. Euler Partition Products
19
for q 0 has zeros precisely at the points and these are first-order zeros. Setting z = 1 and z = -1 in Q(z, q) gives, respectively, the products (1 + q)(1 + q2 )(1 + q3 ) • ...
and (1 - q)(1 - q 2 )(1 - q3 ) • . ,
q E E.
which are holomorphic in the unit disc. As we will see in Subsection 1, their power series about 0 play an important role in the theory of partitions of natural numbers. The expansion of -(f) contains only those monomials qv for which n is a pentagonal 'number 1(31)2 ± v): this is contained in the famous pentagonal number theorem, which we discuss in Subsection 2. In Subsection 3 we expand Q(z . q) in powers of z.
no
1. Partitions of natural numbers and Euler products. Every representation of a natural number n > 1 as a sum of numbers in M{0 } is called a partition of n. The number of partitions of n is denoted by p(n) (where two partitions are considered the same if t hey differ only in the order of their summands); for example, p(4) = 5, since 4 has the representations 4 = 4, 4 = 3+1, 4 = 2+2, 4 = 2+1+1, 4 = 1+1+1+1. We set p(0) := 1. The values of p(n) grow astronomically:
p(n)
7
10
30
50
100
200
15
42
5604
204,226
190,569,292
3,972,999,029,388
In order to study the partition function p. Euler formed the power series Ep(v) q "; he discovered the following surprising result. Theorem ([I]. p. 267). For every q E E,
ac•
(*
ll(i _ g„ ) _1_ )
v-=- 1
Sketch of proof. One considers the geometric series (1 - q") -1 = E:1 0 q"k, q E E, and observes that qv) - = pr,(k)q k , q E E, n > 1, where p7,(0) := 1 and, for n > 1, p(k) denotes the number of partitions of k whose summands are all < n. Since p(k) = p(k) for n > k, the assertion follows by passing to the limit. A detailed proof can be found in [HW1 (p. 275).
FITL, -
There are many formulas analogous to (*). The following appears in Euler ([I], pp. 268-269):
Let u(n) (resp., v(n)), denote the number of partitions of n > 1 into odd (resp., distinct) summands. Then, for every q E E.
Ho. e _1 ) _1
umq„,
+ q") = 1 + v>1.
v(v)e.
v>1
20
1. Infinite Products of Holomorphic Functions
From this, since
1 - g2 1 - g 4
(1 + q)(1 + g2 )(1 + g3 )
1—
q6
1 - q 1 - g 2 1 - g3 • • • 1 1 1 1 - g 1 -
(13
— (1 5 *
one obtains the surprising and by no means obvious conclusion
u(te) = v(n), n > 1.
D
Since Euler's time, every function f : N C is assigned the formal power series F(z) = E f (v)::'; this series converges whenever f (v) does not grow too fast. We call F the generating function of f; the products H(1 - q") -1 , H(1 - q 2v -1 ) and 11(1 + (r) are thus the generating functions of the partition functions p(n), u(n), and 1)(n), respectively. Generating functions play a major role in number theory; cf.. for instance, [1-1W] (p. 274 ff.).
2. Pentagonal number theorem. Recursion formulas for p(n) and cr (n) . The search for the Taylor series of Fil e) about O occupied Euler for years. The answer is given by his famous
_
Pentagonal number theorem. For all g
fi v>1 (*)
- g')
e 1E,
1+ E ( _ 1) f [ 0(3,2_0 1.>1 oc,
E _„_ (1 2 - g 35
—
+ (15
g40
(1 51
q7
q 12
q 15 + (122
q26
+•••.
We will derive this theorem in 5.2 from Jacobi's triple product identity.
- v), which begins with 1, 5, 12, 22, 35, 51, was The sequence u.)(v) already known to the Greeks (cf. [D], p. 1) Pythagoras is said to have determined w(n) by nesting regular pentagons whose edge length increases by 1 at each stage and counting the number of vertices (see Figure 1.1). Because of this construction principle, the numbers ‘,../(v), v E Z. are called pentagonal numbers; this characterization gave the identity (*) its name. Stat ements about the partition function p can be obtained by comparing coefficients in the identity 1 = (Ep(v)e) (1 + E(-1)1(f" v>o v>1
which is clear by 1(*) and (*). In fact, Euler obtained the following formula in this way. (Cf. also [HIM, pp. 285-286.)
§4*. Euler Partition Products
21
• FIGURE 1.1.
Recursion formula for p(n)_ If we set p(n) := 0 for n < O, then p(n) = p(n -1) + p(n, - 2) - p(n - 5) - p(n 7) +... w( -- k))]•
[13 (n w(k))
=
k>1
It was a great surprise for Euler when he recognized — and proved, using the pentagonal number theorem — that almost the sanie formula holds for d denote the sum of all positive divisors sums of divisors. Let a(n) := of the natural number n > 1. Then we have the
E dln
Recursion formula for a(n). If we set a(v) := 0 for v < 0, then a(n) = cr(n - 1) + o-(n - 2) - cr(n, - 5) - a (n - 7) + • • • E(-1)k-1[o-(n - w(k)) + cr(n - w(-k))1 k>1
for every natural number n > 1 that is not a pentagonal number. On the other hand, for every number n = Jj (3v 2 ± 1/), u > 1, cr(n) = (-1)P- in + cr(n - 1) + a(n - 2) - cr(n - 5) - cr(n -- 7) + • • • = (-1)v
+ E(-1)k-i[u(n - w(k)) + a(n k>1
Often, in the literature, only the first formula is given for all n > 1, with the provision that the summand cr(n n), if it occurs, is given the value n. Euler also stated the formula this way. For 12 = (3 • 3 2 — 3), we have —
a(12) = (-1) 2 12+a(11)+a(10)
—
a(7)
—
(5) + cr(0) = 12+12+18-8-6 = 28. -
22
1
Infinite Products of Jlolomorphic Functions
Proof of the recursion formula for er(n) according to Euler One takes the logarithmic derivative of (*) A simple t ransformation gives
(+) 11 r
vg" 1 — (11'
o
= =-
n- I w (
n ) q -0(n)
"SC
The power series about 0 of the first series on the left-hand side is the two series gives a double sum with general terni (-1)()q. Multi)yng Grouping together all terms with the sanie exponent gives
E [EHoka(n — w(k))1q 7' n=1
The assertion follows by comparing coefficients in (±). There appear to be no known elementary proofs of the recursion formula for a(n). The function a(n) can be expressed recut sively by means of the function p(n). For all n > 1,
o- (n) = p(n — 1) -1-2p(n — 2) — 5p(ii -- 5) — 7p(rt — 7)+ 4-12p(n — 12) + 15p(n — 15) — •
w( —k)p(11 —
_ ok-i[u.,(1,- ) p(n — k> I
This was observed in 1884 by C'. Zeller [4 We note anothei formula that can be derived by means of the pentagonal number theorem: 7)
p(n
)
1 - Eu(v)p(n — v).
=
1)=1
3. Series expansion of fr=1 (1
qu z) in powers of z. Although the power series expansion of this function in powers of q is known only for special values of z (cf. Subsections t and 2), its expansion in powers of z can be found easily. If we set Q(z, q) := fl,,, (1 + z), it follows at once that
(1)
(1 + qz)Q(qz,q) = 8
(1 —
Series of the type av e/(1 — q") are called Lambert serzes. Since g" • = g" , the following is immediate (cf. also [Ku]. p. 450).
If the Lambert series
,
°of A ve f, -
— g'') converges normally ni E. then E E, where A,, :=
ad. d
§4*. Euler Partition Products
23
for (q, z) E E x C, this functional equation immediately gives q 2 v(fr+ 1)
OC
11(1
(2)
z) = 1 +
q •, 1 )(
v=.-1
,
.(1 _qv)
- •
Ei,,„
a,,zfr be the Taylor series for (2(z, q). Then Proof. For fixed q E E, let recuision formula ao = 1, and (1) gives the
a,q1' + ay _ ' qv = ay ;
a,, =
i.e.
1 - qv
av-1
for ii > 1.
It follows from this (by induction, for example) that a,, = q is"-F I ) [(I _ For z := 1, we see that
3 (1 + 0(1
q2 )(1
=
q3 )
1- q
+
( 3)
(1 —
(1 - q)(1 - q 2 ) (1 6
q) ( 1 - q2 )(1 - q3 )
If we write q2 instead of q in (2) and set z := C I , we obtain
H (1 + q 2"---1) = 1 + E
2
cx-
(1 20
q4
q 2") .
or, written out,
(1 + q)(1 + q3 )(1 + q5 ) . . = 1 +
q
1 - q2
4
(1 - q2 )(1 - q.1 ) (1 9
(1 - q 2 )(1 - q4 )(1 - (J 6 ) This derivation and more can be found in [I], P. 251 ff.
.±
0
The product Q(z, q) is simpler than the sine product. Not only does normal convergence already follow because of the geometric series, but the functional equation (1), which replaces the duplication formula for s(z), follows easily and is also more fruitful. Exercises. Show that the following hold for all (q. z) E E x C:
a)
II
oc E (1 - (1)(1 — q 2 ) • .. • (1 — q'')" 1 — q"z = 1 + v=i
1
24
1. Infinite Products of Holomorphic Functions q v2
b)
H 1 - q"z
v=1
=1+
_ q)(1 - q2 ) • ... • (1 - q")
(1 - qz)(1 - q 2 z)- ... • (1 - q" z) . Compare the results for z = 1. Hint. For a), first consider [ILI 1 , 1 < n < oo. Find functional equations in each case and imitate the proof of (2); for a), conclude by letting n Do. Equation b) can be found, for example, in the Fundamenta (Pad, pp. 232-233). 4. On the history of partitions and the pentagonal number theorem. As early as 1699, G. W. Leibniz asked Joh. Bernoulli in a letter whether he had studied the function p(n); he commented that this problem was important though not easy (Math. Schriften, ed. Gerhardt, vol. III-2, p. 601). Euler was asked by P. Nauclé, a Berlin mathematician of French origin, in how many ways a given natural number n could be represented as a sum of .s distinct natural numbers. Euler repeatedly considered these and related questions and thus became the father of a new area of analysis, which he called "partitio numerorum." In April 1741, shortly before his departure for Berlin, he had already submitted his first results to the Petersburg Academy ([Eu], I-2, pp. 163-193). At the end of this work he stated the pentagonal number theorem, after he had determined the initial terms of the pentagonal number series up to the summand q51 by multiplying out the first 51 factors of 11(1 - gu) (loc. cit., pp. 191-192). But almost 10 years passed before he could prove the theorem (letter to Goldbach, 9 June 1750; [C], vol. 1, pp. 522-524). In its introduction, Chapter 16 deals thoroughly "with the decomposition of numbers into parts"; the pentagonal number theorem is mentioned and applied (p. 269). The recursion formula for the function p(n) first appears in 1750, in the treatise De Partitione Numerorum ([Eu], I-2, p. 281). It was used in 1918 by P. A. Macmahon to compute p(n) up to n = 200; he found that p(200) = 3, 972,999, 029,388 (Proc. London Math. Soc. (2) 17, 1918; pp. 114-115 in particular). In 1741, Euler had already verified the recursion formula for o(n) numerically for all n < 300 (letter to Goldbach, 1 April 1741; [C], vol. 1, pp. 407-410). In that letter, he called his discovery "a very surprising pattern in the numbers" and wrote that he "would have [no] rigorous proof. But even îf I had none at all, no one could doubt its truth, since this rule is always valid up to over 300." He then informed Goldbach of the derivation of the recursion formula from the (then still unproved) pentagonal number theorem. He gave a complete statement with a proof in 1751, in "Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs" ([Eu], I-2, pp. 241-253). — The reader can find further historical information and commentary in [Weil], pp. 276-281. Not until almost eighty years later could Jacobi give the complete explanation of the Euler identities with his theory of theta functions. We examine this a bit more closely in the next section.
§5*, Jacobi's Product Representation of the Series J(z,q) :=
§5*.
z'
25
Jacobi's Product Representation of the Series
q) := 2
2
The Laurent series E q" 'I/ = + v = q'' (Z 1' + Z -11 converges for every q E E; thus J(z,q) E for all q G E. Readers familiar with the theta function will immediately observe that
ocx )
d(z. r) = .1 (27riz (- ")
( (' f. 1.12.4);
this relation, however, plays no role in what follows. It is immediate that
J(i,q) = I - 1 q4
(1)
q G E.
Jacobi saw in 1829 that his series J(z,q) coincided with the product
A(z,q)
rii(i
_ q2„ )(1 , (12"—
I z )(1
which had been studied by Abel. A(z. E 0(C )( ) for every q E E, since the product converges normally in Cx for each q. The following relation holds between the Euler product Q(z,q) of 4.3 and A(z,q):
z,q 2 ) O(r
A(z.q) =11(1 -
q2) .
The identity J(z,q) = A(z.q), called Jacobi's triple product identity. is one of many deep formulas that appear in Jacobi's Fandaincida Nova. We obtain it in Subsection 1 with the aid of the functional equations
(2) A(q 2 z,q) = (qz) -1 A(z,q),
A(2 -I ,q) = A(z.q),
(z,.q) E C x x Ex .
A(i,q) = A( - 1, q4 ), q E E.
(3)
all of which can easily be deduced from the definition of A; in the proof of (3), we observe that oc
oc
(1 - q2" )
= 11 [(1_ (1,11(
)),
+q 2i/— 1 i)(
_VI , — 1 )
I/=1
Fascinating identities. some of which go back to Euler, result from considering special cases of the equation .1(z, q) = A(z, q); we give samples in Subsection 2. 1. Jacobi's theorem. For all (q, z) E E x C x
q „„ z , =
(J) -
DO
CX,
_, 2„)(i v=- 1
q2"2)(1
q2v—
)
26
1. Infinite Products of Holomorphic Functions
Proof (cf. [HW], pp. 282-283). For every q E E, the product A(z,q) has a Laurent expansion Ec'coo a 1/ 2:Y about 0 in Cx , with coefficients a, that depend on q. Equations (2) of the introduction imply that a_, = a, and a, = q2 u -1 a 11 _i for all uE Z. From this it follows (first inductively for > 0 and then in general) that a, = eaci for all u E Z. It is thus already clear, if we write a(q) for ao , that
A(z , q) = a(q), (z q)
a(0) = 1.
with
A(1, q) and J(1, q) are holomorphic in E as functions of q and J(1,0) = 1; hence a(q) is holomorphic in a neighborhood of zero. From equations (1) and (3) of the introduction it follows, because J (i, q) # 0, that
a(q) = a(q4 )
and hence
a(q) = a(q4 " ), n > 1,
for all q E E.
The continuity of a(q) at 0 forces a(q) = lim„„ a(q4n ) = a(0) = 1 for all E E. The idea of this elegant proof is said to date back to Jacobi (cf. [HAT], p. 296). The reader is advised to look at Kronecker's proof ([Kr], pp. 182-186). With z := e 2"', (J) can be written in the form
q 112 e 2" =
E
[(1 - q 2 ')(1 + 2q2u -1 cos 2w + e -2 )]. u=i
The identity (J) is occasionally also written as OC
(Y
)
00
E (-1)' q 1m(v+1)zu = (1 - z -1 )
- e)(1 - qyz)(1
qv z -1 )].
(J') follows from (J) by substituting -qz for z, rearranging the resulting product, and finally writing q instead of q2 .
2. Discussion of Jacobi's theorem. For z := 1, (J) gives the product representation of the classical theta series OC
E qv = 1 + 2 E
(1)
oc
00 2
2
=
v =1
1.1 =-00
v=1
which converges in E. We also note:
Suppose that k,1 E NVO} are both even
be)th odd. Then, for all (z. q) E
C x x E,
-110V+O z li (2) Eq 2 -00
TT LI v=1
_ qkv)( i fi km-(k.--O z )( 1±grkv---1(k+I)
Z
)j.
2
q' z
§5*. Jacobi's Product Representation of the Series J (z , q) :=
27
Proof. First. let 0 < q < 1. Then qt k ,qi E (0, 1) are uniquely determined,
and substituting ql k for q and ql i z for z turns (.1) into (2). By the hypothesis on k and 1, all the exponents in (2) are integers (!); hence the left- and right-hand sides of (2) are holomorphic functions in q E E for fixed z. The assertion follow g from the identity theorem. For k = I = 1 and z = 1,(2) becomes 00
"0
(3) E q 1 ,(i-f-1) -
+
.2
E v= 1
710
L1
1
this identity, due to Euler, was written by Gauss in 1808 as follows ([Ga ] , p. 20): (3/ ) 1 +
q 3 + q6 + q 10 + etc.
1 _ qq 1 q4 1 q3 1 1— q 1 —
(16
1 (15 1
—
—
q8
q7 etc. 0
(to prove this, use Exercise 2) of 2.1). For k = 3. 1 = 1, and z = —1, equation (2) says that 0C
H[(i _ q 3, )(i _ e v_1 )(1 _ q3,, )1= E Since each factor 1 — qv , y > 1, appears here on the left-hand side exactly once, this yields the pentagonal number theorem 1 +E(
(4)
_ o v {q1 (3v-1)
+ q 1(3v-1-1)i ,
q
E
E,
v=1
v= 1
as announced in 4.2. Written out, this becomes
(1 — q)(1 — q 2 )(1 — q 3 )... 1 q 2 + q5 + (1 7
(4')
(1 12
q 15
+
Now, in principle, the power series about 0 of 1-1 (1+ q") can also be
computed. Since
-x v=1
no
-
q") nu
+ qv) =11(1 — q2"),
we use (4') to obtain
1 — q 2 - q 4 ± q 10 + q 14 - • • • 1 — q — q2 + q5 + — • • • 1 + q + q2 + 2q3 + 2q4 + 3q5 + 4q6 + 5q7 + • • • .
The first coefficients on the right-hand side were already given by Euler; no simple explicit representation of all the coefficients is known. Number-
28
1. Infinite Products of Holomorphic Functions
theoretic interpretations of the formulas above, as well as further identities, can be found in [HW1. 0 We conclude this discussion by noting Jacobi's famous formula for the cube of the Euler product (cf. [Jai], p. 237, and PF1, p. 60):
00 fl (i qV ) 3 =
(5)
Ec-1y(2v-F1)0v( , v.0
+1).
To prove this, Jacobi differentiates the identity (J') of Section 1 with respect to z, then sets z := 1 (the reader should carry out the details, grouping the terms in the series with index v and - 1). In 1848, referring to identity (5), Jacobi wrote ([JF], p. 60): "Dies mag wohl in der Analysis das einzige Beispiel sein, dafi eine Potenz einer Reihe, deren Exponenten eine arithmetische Reihe zweiter Ordnung [. quadratischer Form an 2 + bn + bilden, wieder eine solche Reihe giebt." (This may well be the only example in analysis where a power of a series whose exponents form an arithmetic series of second order [=-- quadratic form an 2 + bn + c] again gives such a series.)
3. On the history of Jacobi's identity. Jacobi proved the triple product identity in 1829, in his great work Fundamenta Nova Theoriae Functionum Ellipticarum; at that time he wrote ([Ja m ], p. 232): Aequationem identicam, quam antecedentibus comprobatum ivimus: (1 - 2q cos 2x + q2 ) (1 - 2q 3 cos 2x + q6 ) (1 - 2q 5 cos 2x + q 1°) . . . 1- 2q cos 2x + 2q4 cos 4x - 2q9 cos 6x + 2q 16 cos 8x (1 - q2 )(1 - q4 )(1 - q6 )(1 - q8 )... In a paper published in 1848, Jacobi systematically exploited his equation and wrote (Pa21, p. 221): Die siimmtlichen diesen Untersuchungen zum Grunde gelegten Entwicklungen sind particuldre Fiille einer Pundamentalformel der Theorie der elliptischen Rinctionen, welche in der Gleichung ( 1 _ g2)( 1 _ q4)( 1 _ q6)( 1 _ qs) x(1 - qz)(1 - q3 z)(1 - q5 z)(1 - q7 z)...
x(1
-
qz -1 )(1 - q3 z -1 )(1- q5 z -1 )(1 -
1 _ q( z z -1)
q4( z 2
z -2) _ q9( z 3
z -3)
enthalten ist. (All the developments that underlie these investigations are special cases of a fundamental formula of the theory of elliptic functions, which is contained in the equation ....)
§5* • Jacobi's Product Representation of the Series .1(z, q) :=
_ ql z' •
29
The preliminary work for the Jacobi formula was carried out by Euler through his pentagonal number theorem. In 1848 Jacobi wrote to the secretary of the Petersburg Academy, P. H. von Fuss (1797 1855) (cf. 1.1F1, p.60): "Ich m6chte mir bel dieser Gelegenheit noch erlattben, Ihnen zu sagen. warum icli mich so fiir diese Eulersche Entdeckung interessiere. Sic ist niimlich der erste Fall gewesen, in welchem Reihen aufgetreten sind, deren Exponenten eine alit hMetische Reihe zweiter Ordnung bilden, und auf diese Reihen ist durch mich die Theorie der elliptischen Transcendenten gegriindet worden. Die EUlersche Formel 1st ein specieller Fall enter Formel, welche wohl das wichtigste und fruchtbarste ist, was ich in miner Mathematik erfunden habe." (I would also like to take this opportunity to tell you why T am so interested in Euler's discovery. It, was, you see. the first case where series appeared whose exponents form an arithmetic series of second order, and these series, through my work, form t he basis of the theory of elliptic transcendental functions. The Euler formula is a special case of a formula that is probably the most important and fruitful I have discovered in pure mathematics.) Jacobi did not know that. long before Euler, Jacob Bernoulli and Leibniz had already come across series whose exponents form a series of second order. In 1685 Jacob Bernoulli, in the Journal des Sravans, posed a problem in probability theory whose solution he gave in 1690 in Arta Erudztorum: series appear there whose exponents are explicitly asserted to be arithmetic series of second order. Short ly also solved the problem; he after Bernoulli, Leibniz — in Acta Erud,itorum considered the question especially interesting because it might lead In series that had not yet been thoroughly studied (ad series tamen non sat is adhuc examinatas ducit). For further details, see the article [En] of G. E. Enestriim. In his Ars Conoctandi, Bernoulli returned to t he problem; I he series - 771
1 t/ m– m+ m– zn +
21
-
28
+
.46
-15
–m+•••
appears in [B] (p. 142). Bernoulli says that he cannot sum the series but that one can easily "compute approximate values to arbitrarily prescribed accuracy" (from R. Haussner's German translation of [B], p. 59). Bernoulli gives I he approximation 0.52393, which is accurate up to a unit in the last decimal place.
Gauss informed Jacobi that he had already known this formula by about 1808: cf. the first letter from Jacobi to Legendry (PLI, p. 394). Legendre, bitter toward Gauss because of the reciprocity law and the method of least squares, writes to Jacobi on the subject ([JL], p. 398): - Comment se fait-il que M. Gauss ait osé vous faire dire que la plupart de vos t héorèmes l ui était connus et qu'il en avait, fait la découverte dès 1808? Cet excès d'impudence n'est pas croyable de la part d'un homme qui a assez de mérite personnel pour n'avoir besoin de s'approprier les découvertes des autres ." (How could Mr. Gauss have dared inform you that most of your theorems were known to him and that, he had discovered them as early as 1808? Such outrageous impudence is incredible in a man with enough ability of his own that he shouldn't have to take credit for other people's discoveries ....)
1. Infinite Products of Holomorphic Functions
30
But Gauss was right: Jacobi's fundamental formula and more were found in the papers he left behind. Gauss's manuscripts were printed in 1876, in the t bird volume of his Werke: on page 440 (without any statements about convergence) is the formula (1+ sy)(1+ x3 y)(1 + x 5 y) ... (1. +
=
{1 + (y +
+
)
+
+
(yy + — uly ) + .r9 (y3 +
+•
,
where [x.r] stands for (1 — x2 )(1 — x 4 )(1 — x 6 ) .... This does in fact give Jacobi's result (J). Schering, the editor of this volume, declares on page 494 I hat this research of Gauss probably belongs to the year 1808. Kronecker, generally sparing of praise, paid tribute to the triple product identity as follows ([Kr], p. 186): "Hierin besteht die ungeheure Entdeckung Jacobi's; die Umwandlung der Reihe in chas Produkt war sehr schwierig. Abel hat auch das Produkt, aber nicht die Reihe. Deshalb wollte Dirichlet sic auch ais Jacobi'sche Reihe bezeichnen." (Jacobi's tremendous discovery consists of this: the transformation of the series into the product was very difficult. Abel too had the product, but not the series. This is why Dirichlet also wanted it to be called the Jacobi series.) The Jacobi formulas are only the tip of an iceberg of fascinating identities. In 1929, 0. N. Watson ([Wa], pp. 44 45) discovered the Quintuple product identity. For all (q, z) E E x Cx ,
E =
3s,
+ z
—,iv
-2
—z
X,
fl ( 1
1 2v )( 1
1 2"— 1
z )( 1
q 2v —1 z — 1 )(1
(7 4t1 -
z2
— 2) .
v= I
Many additional formulas come from considering special cases: see also [Go] and [Ew]. For some years there has been a renaissance of the Jacobi identities in the theory of affine root systems. As a result. identities have been discovered that were unknown in the classical theory. E. Neher, in [N], gives an introduction with many references to the literature.
Bibliography 13EliNouLLI, .1.: Ars Conjeetandi, Die Werke von Jakob Bernoulli, vol. 3,
107 -286, Birkhijuser, 1975; German translation by R. HA USSNER Klassiker 107 (1895).
, OSWaid'S
Von Fuss, P..1. (ed.): Correspondance inattimatique et physique de quelques ee'lèbres Vomètres du X VIIi d'me szcle, St Petersburg, 1843. 2 vols.; reprinted in 1968 by Johnson Reprint Corp.
a
l acobi's Product Representation of the Series .1(z,q):=
q" 2 z"
31
ID]
DICKSON, L. E.: History of the Theory of Numbers, vol. 2, Chelsea Publ. Co., New York, 1952.
I Eu
EISENSTEIN, U. G. NI.: Genaue IIntersuchung der undendlichen Doppelproducte, ans welchen die elliptischen Functionen ais Quotienten zusammengesetzf sind, und der mit ihnen zusammenlningenden Doppelreihen, Journ. reine angew. Math. 35, 153 274 (1847); Math. Werke 1, :357 478.
[En]
ENESTRÔNI.
[Eu]
Leonhardi EULERI Opera oinnia, sub auspiciis societatis scientarium naturalium Helveticae, Series 1-IV A, 1911-.
[Ew]
EwELL, .1. A.: Consequences of Watson's quintuple-product identity, Fibonacci Quarterly 20(3), 256-262 (1982).
[Ga]
GAt'ss, C. F.: Summatio quarundam serierum singularium, Werke 2, 9 45.
[Go]
Con DON, B.: Some identities in combinatorial analysis, The Quart. Journ.
G. E.: Jacob Bernoulli und die Jacobischen Thetafunction. Bibi. Math 9, 3. Folge, 206-210 (1908-1909).
Math. (Oxford) 12, 285-290 (1961). [HW]
HARDY. G. H. and E. M. WRIGHT: An Introduction to the Theory of Numbers. 4th ed., Oxford. Clarendon Press, 1960.
[I]
Eul.KH. L.: Introduetio w Analysin Infinitoruin, vol. 1, Lausanne 1748; in l Eu, 1-8; German translation, Einleitung in der Analysis des linendlichen, 1885, published by Julius Springer; reprinted by Springer, 1983.
[Jar]
JAcom, C. G. J.: Fundamenta nova theoriae functionum ellipticarum. Ges. Wk e 1. 49-239.
[Ja2j
JAcorn. C. G. .1.: Veber unendliche Reihen, deren Exponenten ingleich in zwei verschiedenen quadratischen Formen enthalten sind, Journ. mine angew. Math. 37, 61 94 and 221- 254 (1848), Ces. Welke 2, 217 288.
[JF]
C .1. Jacobi .1 A('0131, C. G. J. and P H. von Fuss: Briefireclisel zwischen und P. II. von Fuss iiber the Herausgabe der Werk.e Leon/aid Eiders. ed. P. SiAcKEL and W. AHRENS, Teubner. Leipzig, 1908.
PLI
JAcout, C. G. .1.: Correspondance mathematique aver Legendre, Werke 1, 385 461.
[Kn]
KNOPP. K. Theory and Application of Infinite Series. Heffner Publishing Co., New York, 1947 (trans. R. C. 11. YouNG).
[Kr]
KRONECKER, L: Theorie der cigar:hen und der inelfachcn !Wed( late, ed. E. NE'TTO. Teubner. Leipzig, 1894.
[M]
MooRE, E. H.: Concerning the definition lw a system of functional properties of the function f(z) = sin 7r)/7r. Ann. Math. 9. 1st ser., 43 49 (1894).
Ces.
(
[N]
Jacobi's Tripelprodukt Identitat und N-Identitaten in der Theone affiner Lie-Algebren, Jber. DM V 87, 164 181 (1985).
[Nu]
Numbers. Springer-Verlag, New York. 1993; ed. .1. FI. EWING; trans. H. L. S. ORDE.
[11
PRINGSHEINI. A.: -Ober die Convergenz unendlicher Produkte, Math. Ann. 33, 119-154 (1889).
NEHER,
32
1. Infinite Products of Holomorphic Functions
[H
R irr. .1. F.: Representation of analytic functions as infinite products, Math. Zeitschr. 32, 1--3 (1930).
[V]
VA LI RON • G.: Thorie des fonctions, 2nd ed., Masson. Paris. 1948.
[Wa]
WATSON , G. N.: Theorems stated by Ramanujan (VII): Theorems on continued fractions„Journ. London Math. Soc. 4, 39-48 (1929).
[Wei]
K. I:ber die Theorie der analytischen Facultaten. Journ. mine angry,. Math. 31, 1 60 (1856): Math. Werke 1, 153-221.
[Weill
WEiL, A.: Number Theory: An Approach through lbstory, from Hammurape to Legendre. Birkluinser, 1984. ZEi.txtt., C.. Zu Eulers Recursionsformel fiir die Divisorensummen, Acta Math. 4. 415 116 (1884).
2 The Gamma Function
Also das Product 1 2 • 3 ... .. it die Function, die meiner Nleinung int,ch in der Analyse eingefiihrt werden muss. (Thus the product 1 • 2 • 3 ....t is the function that, in my opinion, must be introduced into analysis.) C. F. Gauss to F. W. Bessel. 21 November 1811
1. The problem of extending the function n! to real arguments and finding the simplest possible "factorial function' . with value n! at n E N led Euler in 1729 to the r-function. He gave the infinite product :.‹. , I 1• 2' 2 1- z3z 3'4= 1 r(z + 1) : = ... = H (i ,_ _) (, _,_ --,) v v , ._,_ z 2 + z 3+ --,. as a solution. Euler considered only real arguments; Gauss, in 1811, admitted complex numbers as well. On 21 November 1811, he wrote to Bessel (1784-1846), who was also concerned with the problem of general factorials, "Will man sich aber nicht ... zahllosen Paralogismen und Paradoxen und Widerspriichen blossstellen, so muss 1.2-3 X nicht ais Definition von H j . gebraucht werden, da eine solche mur, wenn r eine gauze Zahl ist, einen bestimmten Sinn hat, sondern man muss von einer haeren allgemein, selbst 'Precise references to Euler can be found in the appropriate sections of this chapter; we rely to a large extent on the article "tibersicht fiber die Bande 17, 18. 19 der ersten Serie" of A. Krazer and G. Faber in [Eu[, 1-19, pp. XLVII I,XV in particular.
3-1
2. The Gamma Function
auf iniaginre Werthe von x anwendbaren, Definition ausgehen, vvovon jene ais specieller Fall erscheint. Ich habe folgenden gewdhlt H.r =
1.2.3.. x + 1.x + 2..r + 3 ... x +
wenn k unendlich win!." (But if one doesn't, want ... countless fallacies and paradoxes and contradictions to be exposed, 1 • 2 • 3 ... x must not be used as 11w definition of x, since such a definition has a precise meaning only when x is an integer: rather, one must start with a definition of greater generality, applicable even to imaginary values of x, of which that one occurs as a special case. I have chosen the following ... when k becomes infinite.) (Cf. [Gd, pp. 362-363.) We will understand in §2.1 why, in fact. Gauss had no other choice. The functions of Euler and Gauss are linked by the equations
ri
F(z + 1) = II(z), F(n + 1) = 11(n) = n!
for n = 1,2,3, ....
The F-funct ion is merarrimplute ni C: all its poles are of first order and occur at the points -n, n E N. This function has the value n! at n +1 (rather than n) for purely historical reasons. Gauss's notation Hz did not last. Legendre introduced the now-standard notation F(z) in place of II(z - 1) (cf. [L 1 ], vol. 2, p. 5); since then, one speaks of the gamma function. 2. In 1854, \Veierstrass made the reciprocal 1
Fc(z) := r(z) •
\z
_ll( v+i ) 0+_)=4-10++-z1og(-1-)
of the Euler product the starting point for the theory; Fc(z), in contrast to F(z), is holomorphic everywhere in C. Weierstrass says of his product, ((Wed, p. 161): "Ich meichte ffir dasselbe die Benennung 'Factorielle von u' und die Bezeichnung Fe(u) vorschlagen. indem die Anwendung dieser Function in der Theorie der Facultiiten dem Gebrauch der F-Function deshalb vorzuziehen sein darfte, veil sie fur keinen Wert von u eine linterbrechung der Stetigkeit erleidet und iiberhaupt im Wesentlichen den Charakter einer rationalen ganzen Function besitzt." (I would like to propose the naine "Fa('torielle of a" and the notation Fc(u) for it, since the application of this function in the theory of factorials is surely preferable to the use of the F-function because it suffers no break in continuity for any value of a and, overall, ...essentially has the character of a rational entire function.) Moreover, Weierstrass almost apologized for his interest in the function Fe(u); he writes (p. 158) Theorie der analytischen Facultiiten in meinen Augen durchaus nicht, die Wichtigkeit hat, die ihr in friiherer Zeit viele Mathematiker beimassen" (that the theory of analytic factorials, in my opinion, does not by any means have the importance that many mathematicians used to attribute to it).
2
The Gamma Function
35
Weierstrass's "Factorielle" Fe is now usually written in the form zelz
H
:= lim
e- ,
E" _1 _ ion )= Enler's constant.
We set A := Fe and compile a list of the most important properties of A in Section 1. The F-function is studied in Section 2. Wieland's uniqueness theorem, which, for example, immediately yields Gauss's multiplication formula, is central. 3. A theory of the gamma function is incomplet e without classical integral formulas and Stirling's formula. Euler was familiar with integral representations from the outset: the equation n! =
o (- log x) d.r, n G N, •
appears in his first work on the F-function, in 1729. For a long time, Euler's identity F(z) =
tz -l e -i dt
for z E C.
Re z > O.
has played the central role; in Section 3 we derive it and Hankel's formulas by using Wielandt's theorem. In Section 4 Stirling's formula, with a universal estimate of the error function, is also derived by means of Wielandt's theorem; at the same time, following the example of Stieltjes (1889), the error function is defined by an improper integral. In Section 5, again using the uniqueness theorem. we prove that B(n), z) =--
tw -1 .
0
- I ) 2- dt 1"(1/1
Textbooks on the F-function: [A]
ARTIN, E.: The Gamma Function, trans. NI. Bt - TLER, Holt, Rinehart and Winston, New York. 1964.
[Li]
LINDEL6•, E.: Le calcul des résidus, Paris, 1905; reprinted 1947 by Chelsea Pub!. Co., New York; Chap. IV in particular. NIELSEN, N.: Handbuch der Th,eorie der Gaminafunktion, first printing Leipzig, 1906; reprinted 1965 by Chelsea Publ. Co.. New York.
[WW]
WHITTAKER. E. T. and G. N. WATSON: A Course of Modern Analysis, 4th ed., Cambridge Univ. Press. 1927, Chap. XII in part icular.
L. W. V. Another instructive reference is an encyclopedic art irk by JENSEN: An elementary exposition of the theory of the gamma function, Ann. Math. 17 (2nd ser.), 1915 16, 124-166. For the reader's convenience, these references are listed again in the bibliography at the end of this chapter.
36
2. The Cantina Function
§1.
The Weierstrass Function A(z) -= ze"fz n v>i (1 + z/v)e - z/v
In thii section we collect basic properties of the function A. including
A E 0(C), A(z) = zA(z + 1). 7rA(1 - z) = sin 7rz. 1. The auxiliary function result is fundamental.
(1) The product
1-1„, 1 (1 +
H (z) := z11,7
+
z1v)e - z/z). The next
z1v)e - zlv converges normally in C.
Proof. Let B := B,, (0), n E N\{0}. It suffices to show that
< ou
1-
,,>,
for all n > 1.
-8 71
In the identity tt ,2 [( i _
/1
1\
-
1
(vI.+ 1), )
+
w+••• w+•
all expressions in parentheses on the right-hand side (... ) are positive. Hence
— — w)e" . 1 5_
E
1
1)!
v!
= w 2 whenever w < 1.
For w = -z/v, it follows that 11- (1+ z1v)e -z1v 1 n
E
—
Convergence is produced in the preceding expression by inserting the exponential factor exp(-z/v) into the divergent product 11, >1 (1 + z/v). 'Weierstrass was the first to recognize the importance of this Crick. He developed a general theory from it; see Chapter 3. Because of (1). H (z) := 41(1 + z / v)c - 'I' is an entire function. By 1.2.2. H has zeros, each of first order, precisely at the points -n., n E N. The identity
(2)
- H (z)H(- z) = z2
_ z2,2) = ir1 zsinirz
v>
(I +
§1. The Weierstrass Function A(z) = ze-tz
37
z/v)e - '/'
follows immediately; it says that H(z) consists essentially of - half the factors of the sine product." Furthermore. (3) H(1) = e
with -y 71 —. DC
Proof. Since fl
(1+
=
1 1 1 (1 + - + - +... - - log n) E R. 2 3
n + 1, we have
1 1 H(1) = hum(l -I- - exp(-- = inn ) exp (log(n + 1) /I
1
V
J 0; hence -y := -log H(1) = lim„, - log(n + 1)) E R. Since log(n + 1) - log n =- log (1 + and lim„, log (1 + ;1;.) o. the assertion follows. The real number -y is called Euler's constant;
= 0.5772156....
Euler introduced this number in 1734 and computed it to 6 decimal places gul, 1-14, p. 94): in 1781 he gave it to 16 decimal places ([Eu}, 1-15, p. 115), of which the first 15 are correct. It is not known whether -y is rational or trnitional, nor has anyone yet succeeded in finding a representation for -y with simple arithmetic formation rules like those known, for example, for e and it.
With nz := e z tog n we have (2.„
+
z
4_
0 log u
-
E v=1
thus H can also be written as follows: (4)
H(z)= e --r z lim
+n)
z(z +
n
n !Ti
In the next subsection. the annoying factor product.
is interwoven with the
Exercise (Pringsheim 1915). Let p,q C N\{0}. Prove that
ri 0 _ ..i.•) 11 0 + _z) _v . [exp (z qn
lini It
OC,
[ pn
v=1
v
v=1
v
log, 11(j ) 1
sin nz
zE Cx
Hint. Prove. among other things. that lim n —, Ep1 :4 I 1,, = log for q > P.
2. The entire function A(z) := e11(z) has zeros. all of first order, precisely at the points -n, u E lJ. We have A(z) =
A(.r) > 0
for every .r E R. r > O.
38
2. The Gamma Function
It follows from 1(3) and 1(4) that (1)
A(1) = 1, A(z) = urn
z(z + 1) . (z + •
n !n z
71
From this, since lim(z + n + 1)/n = I. we immediately obtain the Functional equation. A(z) = zA(z + 1).
The sine function and the function A are linked by the equation (2)
rA(z)A( 1 - z) = sin rz.
Proof. This is clear by 1(2), since A(z)A(1 -
= -z -1 A(z)A(-z) =
In 2.5 we will need the multiplication formula (3)
(27 0 (A _ I)A ( i_,I) A.:2) ...A ( k
1) k )
= /7.
h
for k = 2,3, ....
Proof. We use the well-known equation
n k—
(*
2k
)
sin - r = k. k
e -2") and (The quickest way to see this is to observe that sin z = (2i) - e"(1 n k-1 Inr:/k tr(k-1)/2 , write the sine product in (*) in the form =c =7 —
( 21 )1-k 1 k- 1 11(1
_
=
and use the identity 1+ w+ • • • + for w := 1.)
Since k—1
FI AK=1
A(K/k) = 2
k
—
/pis- I
= (w k — 1)/(w— 1) = Fr k-1 ( 11:— P -2 "TKIk
k- 1
, A(1 -
k) holds trivially, (2) and (*) yield
1
k—1
=
H - sin = (27 )k-1 . K
Since A (x) > O for x > O. the assertion follows by taking roots. Exerezse (Weierstrass, 1876). Show that
v>1
v+
)
Y
(,±
§2. The Gamma Function
§2.
39
The Gamma Function
We define
F(z) := 1/A(z) and translate the results of the preceding section into statements alunit the gamma function, thus giving its theory a purely multiplicative foundation.
1. Properties of the l'-function. Our first result is immediate. F(z) is h,olamorphic and nonvanishing in CVO, —1,-2_1: every point —n, n E N. is a first-order pole of F(z). Moreover, (F)
F(z + 1) = zr(z),
with
no = -I
(functional equation).
The functional equation (F) is central to the whole broader theory. For instance, if F(z) is known in the strip 0 < Re z < 1. then (F) can immediately be used to find its values in the adjacent strip 1 < Re z < 2. and so on. In general, it follows inductively from (F), for n E NVOI, that
(1)
F(z + n) = z(z + 1) ... (z + n — 1)F(z),
F(n) = (n — 1)!.
We immediately determine the residues of the gamma function:
res_„F =
(2)
( -1
)"
n!
n E N.
Proof. Since —n is a first-order pole of F, we know that res _„ = n)r (z) (see. for example. 1.13.1.2). By (1).
resF =lim
l'(z + ii + 1) Z(Z + 1) ... (:C: + n — 1)
(-1)"
F(1) (—n)(—n + 1) ... (-1)
n! •
»t(C) that satisfies the equation h(z -I- 1) = Remark. Every function h(z) zh(z) with h(1) E Cx has a first-order pole at each —n. n E N. with residue (-1)"h(1)(n!) -1 . The formula 1.2(1) for A(z) becomes Gauss's product representation:
F(z) = lim
(G)
ri!itz z(z + 1) ... (z +
Plausibility argument that (G) is the "only" equation for function s f that satisfy (F): By (F). for all z, n E N,
f (z + n) =
— 1)!n(n + 1) •
(n — 1)!zt z
+ —1 ) )7
(n + z — 1)
+... • T1
+
40
2. The Gamma Function
f (z + n)I (n — 1)!nz for large n; more precisely, ((n — I)!nz) = 1. If one postulates this asymptotic behavior for arbitrary z, then (1) forces
Clearly f (z + a)
f (z) = lim
f (z + n)
(z + n — 1)
z(z + 1)
(n — 1)!nz z(z + 1)...(z +n — 1) •
= uni 71
which, since lim n/(z + n) = 1, is just Gauss's equation (G). See also Subsection
4 It follows immediately from (1) and (G) that (3)
lui
n
F(z + n) F(n)nz
•
Formula 1.2(2) can be rewritten as Euler 's supplement: (E)
1'(z)F(1 — z) =
77
siri7rz •
It follows immediately from the definition of F(z) that
r(z) = r i)
for x > 0.
and F(x) >
(
Since n = irr and lz + vj > x + u for all z with x = Re z > 0, (G) implies that F(s)
(4)
for all z E C with x = Re z >0.
In particular, F(z) is bounded in every strip { zEC:r 0 (cf. 1(4)), F is bounded in {z E C : 1 < Re z < 2}. By the uniqueness theorem of the preceding subsection, it follows that F = r; hence F(kz) = r(kz), i.e. (1). Historical note. By about 1776, Euler already knew the formulas
(1')
„/ir
(1) r (_2) ...r (k )
)
-1)
(27)1(k-i)
k
([Eu], 1-19, p. 483); they generalize the equation F() = OF. The equations (1) were proved by Gauss in 1812 ([G2], p. 150); E. E. Kummer gave another proof in 1847 [Ku]. Logarithmic differentiation turns (1) into the convenient Summation formula: .0(kz) = log k +-1-1,.E",11=.01 11)(z +
k = 2, 3, ....
For k = 2, (1) becomes the Duplication formula: Nfir(2Z) = 2 2z- 1 r(Z)r(Z
which was already stated by Legendre in 1811 ([Li], vol. 1, p. 284).
46
2. The Gamma Function The identities (1) contain
Multiplication formulas for sin irz. For all
2 +k
1 sin kirz = 2 k-1 sin 7TZ sin ir (z + - ) sin
Proof. Since 1 -
k G N, k > 2, . sin ir (z + k
k
1)
kz = k(-z +1Ik), Euler's formulas 1(E) and (1) yield
7r(sin kirz) -1 =P(kz)F(k(-z + 1/k))
___ (2 7
k-1 )1—k
[r
(z +)r -z +
.
rc=0
It is clear that Fr k-2
r f-z
k
n k-1 P (1 - z - 1'k )• hence 11pc=1 e k—i
+
r (1 -
[7r (sin 7r
+ ))
7r(sink7rz) -1 = (27) 1-k r(z)1- (1 - z)1-I
[r
+ Z))1
ts=1 —1
= (270 1- k 7r(sin rz) -1
.
rc=1
The duplication formula leads to another Uniqueness theorem. Suppose that F E M(C) is positive in (0, cc) and satisfies F(z +1) = zF(z) and \/F(2z) = 22'1 F(z)F(z + Then
F = F.
Proof. For g := F/r E M(C), we have g(2z) = g(z)g(z + 1) and g(z +1) = g(z). Therefore g(x) > 0 for all x E R. Hence, by Lemma 1.3.2, g(z) = aebz , where b is now real. Since g has period 1, it follows that b = 0; hence g(z) 1, i.e. F = F. Exercises. Prove the following:
1) f log r(t)dt = log V'
directly, using the duplication formula (cf. 1.4).
2) f log r(( + z)d( = log 'N,/. 7r. + z log z - z for z E
0) (Raabe's func-
tion). 1
1
1
1
k-1
, k = 2, 3, .... • 6*. Holder's theorem. One can ask whether the r-function — by analogy with the functions exp z and cos z, sin z — satisfies a simple differential equation. O. Holder proved that this is not the case ([HO], 1886). k
(1 +
)
§2. The Gamma Function
47
Holder's theorem. Ther-function does not satisfy any algebraic differential equation. In other words, there is no polynomial F(X, X 0 , X i ,. , X n ) 0 in finitely many indeterm,inates over C such that
F(z,r(z),r (z),
,r( n ) (z))
O.
Weierstrass assigned the proof of this theorem as an exercise. There is a series of proofs, for example those of Moore (1897), F. Hausdorff (1925), and A. Ostrowski (1919 and 1925) (cf. [M], [Ha], and [0]). Ostrowski's 1925 proof is considered especially simple; it can be found, among other places, in [Bi), pp. 356-359. All the proofs construct a contradiction between the functional equation F(z + 1) = zf(z) and the hypothesized differential equation.
7*. The logarithm of the F-function. Since r(z) has no zeros in the star-shaped domain C - , the function ii)(z) = ["(z)/F(z) is holomorphic there and (1)
1(z) := f [1,z)
dC,
z E C - . with 1(1) = 0,
as an antiderivative of the logarithmic derivative of f(z), is a logarithm of F(z) (that is, el( z ) = r(z); cf. 1.9.3). We write log r(z) for the function 1(z); this notation, however, does not mean that 1(z) is obtained in Cloy substituting F(z) into the function log z. It follows easily from (1) and Proposition 3 that 00
(1')
logF(z) =
- log z +
-
log (1+
,
Z E C.
v=1 Proof. Since the partial fraction series -y - 1/( [1/(( + v) - 1/v] converges normally in C- , it can be integrated term by term; cf. 1.8.4.4. For z E C- and v > 1, 1 + z/v E C- and hence log(z + v) - log(1 + v) = log (1 + z/v) - log(1 + 1/v) (!). Thus
00
logF(z) =
E [log(z + v) - log(1 + v) v=i _ log (1 + zj- ) + log (1 + v
--yz + - log z --yz+-y+ v=i
_ vz lv
-
1.
Since En, i [1/v - log (1 + 1/v)] = E vn =i 1/v - log(n + 1) tends to y, (1') E] follows.
48
2. The Gamma Function
We now consider the function log F(z + 1) in E. Its Taylor series about 0 has radius of convergence 1; we claim that
\ (2)
oc
log F(z + 1) = --yz +
Ç(n)z,
where
((n) := v=1
n=2
1 —. // 71
Proof. Since
1 z+v
1 v
1 v [ 1 +1z/v
v n (vz) 1 ] = it •=1 (-1)
if z I < v,
it follows from 3(1) and Proposition 3 that
V.qz
E (t (—ir zn = —,.y +E(-1)fl(rt)z--',
-I- 1) =
, 0+1 Vt=1
ZE
E.
n=2
But (log ['(z + 1))' = Ip(z + 1) and log
r(l) =
0; this gives (2).
For z = 1, the series (2) gives the formula oc,
(Euler, 1769).
(3) rt=2
Proof. Since ((n -I- 1) < ((n), the terms of the alternating series on the right-hand side tend monotonically to 0: hence the series is convergent. Abel's limit theorem can he applied to (2):
( 1 n=2
n
((n) = lim x/1
n=2
n
)"n ((n)xn- = -y + log r(2)
=
Historical note. Rapidly convergent series for log F(z + 1) can be obtained from the series (2); cf., for example, [Ni], p. 38. These give enough information to tabulate the initial values ((n) of the logarithm of the gamma function. Legendre established the first such table: it contains the values of log F(x + 1) from x = 0 to x = 0.5, with increment 0.005, up to seven decimal places. Legendre later published tables from x = 0 to x = 1 with increment 0.001, correct to seven decimal places ([Li], vol. 1, pp. 302-306); in 1817 he improved these tables to twelve decimal places ([L 1], vol. 2, pp. 85-95). Gauss, in 1812, gave the functional values of + x) and log r(1 + x) from x = 0 to x = 1, with increment 0.01, up to twenty decimal places ([G2], pp. 161-162). Euler announced equation (3) in 1769 ([Eu], I-15, p. 119).
*(1
§3.. Euler's and Hankel's Integral Representations of r(z)
49
§3. Euler's and Hankel's Integral Representations of F(z) Euler observed as early as 1729 — in his first. work ([Eu], 1-14, pp. 1-24) on the gamma function - that the sequence of factorials 1, 2, 6. 24, ... is given by the integral
71.! = f (- log 7-)nc/T. n EN (loc. cit., p. 12). In general,
F (z + 1) = f (- log r)zdr whenever Re z > -1; with z instead of z + 1 and t := - log 7- , this yields the equation (I)
F(z) =
tz -l e -t dt, z E T := {z E C : Re z > 0 } .
The improper integral on the right-hand side of (1) was called Euler's integral of the second kind by Legendre in 1811 ([L i ], vol. 1, p. 221). Its existence is not obvious; we prove in Subsection 1 that it converges and is holomorphic. The identity (1) is a cornerstone of the theory of the gamma function; we prove it in Subsection 2. using the uniqueness theormn 2.4. In Subsection 4 we use the uniqueness theorem to obtain Hankel's formulas for F(z). Integral representations of the F-function have repeatedly at t racted the interest of mathematicians since Euler. R. Dedekind obtained his doctorate in 1852 with a paper entitled "fiber die Elemente der Theorie der Euler scheri (cf. his Ccs. Math. Werke 1, pp. 1-31), and H. Hankel qualified as a university lecturer in 1863 in Leipzig with a paper called "Die Eulerschen Integrale bei unbeschninkter Variabilitiit des Argumentes"; cf.
1. Convergence of Euler's integral. We recall the following result. Majorant criterion. Let y : D x[a,x,) C be continuous. where D C C is a region and a E R. Suppose there exists a function (t) on la. c.x.)) such that
M(t) for all z E D, t > a,
and
AI (1)dt E R
Then f: g(z, t)dt converges uniformly and absolutely iii D. If g(z t) E 0(D) for every t > a, then g(z,t)dt is holomorphic in D.
r
2. Tile Gamma Function Proof. Let E > O. Choose b> u such that r Al(t)dt < E. Then •f
t)dt
(l)dt < E for all z E D and e > b.
g(;s: b
b
Tlie uniform and absolute convergence of the integral in D follows from Cauch ■,.'s convergence criterion. If g is always holomorphic in D for fixed then L s g(z.t)dt E 0(D) for all r and s such that o < r < s < oc (cf. 1.8.2.2). Then we also have rc g(z.t)dt E 0(D). (Incidentally. it is easier to show that this integral is holomorphic by using Vitali's theorem: cf. 7.•1.2.) For r E it, let S,1 (resp.„V,,- ) denote the right half-plane Re z > r (resp.,
the left half-plane He z < r). For brevity, we set. -x, u(:):
v(L):= .10
Convergence theorem The integral t, (z) converges uniformly and absolutely iii S, for every r E IR, moreover. v(z) E 0(0). The integral u(z) ronveiges uniformly and absolutely in S 7±. for every r > U. Morcoecr. u(z) E 0(7) and
(
( - 1)v
u(z) =
1 )
1,
I
for every z G T.
z v
=II
Proof. a) For all .-.,- E S,- , we have Itz -1 < tr - I. Since limf—x, tr-te-lt there exists an ill > 0 such that t e < 'It for all z E Si:, t > 1. E 0(0) for all t > 1, the claims tdt = 2/ \/T. and tz Since 1 about v follow from the majorant criterion. b) Set s 1/1; then u(z) ( -I lss - z - ids. If r > O. then = The majorant and moreover .r; < -I for all z E criterion now gives all the claims about u except for equation (1). This follows from tlw identity ( - 1)" 11
1
tz" -I dt 1,0 (-1)" L-s 1,-()
1 z v
which holds for all 6 E (0. 1) (theorem on interchanging the order of integration and summation; cf. 1.6.2.3), since liez > 0 and the last surnmand
§3.. Euler's and Hankers Integral Representations of F ( .-.;) The integrals u(r), r < 0, diverge. Since /'
It
>e
51
in (0, 1),
1
f
tr
dt> e -
dt
(1 - n');
thus lini
= .6
2. Euler's theorem. The zntegral
tz -l e -i dt converges unlfarraly and absolutely to F(z) m every strip {z E C : a < Re z < b}, < a < b < •-)c-: I r -1 di for z E T.
F(z) =
Proof Convergence follows from the convergence theorem I, since t he integral coincides with F := a + e in T. For F tE 0(T), it is immediate that
fqz +
= zF(z).
F(1) =1,
for all z
IF('-'') 1‹ inn(' z-H.
In particular, F is bounded if 1 < Re z < 2. That F =Theorem 2.4.
e
T.
follows from
Of course, there are also direct proofs of the equation F = r The reader may consult. for instance, [A], where the logarithmic convexity of 1" (.1.), > 0. is proved, or [WW1. where Gauss's proof is given. One verifies the equations ri.n t Z
n)
=
I
t :4-1 (1 - 1/11)(11,
./1)
z E T, n = 1,2, ...
by induction, then proves that the sequence on the right-hand si(le converges
The F-integral can be used to determine a number of integrals. The Gaussian error integral, discussed at length in Volume I, is a special I'value: • "X.
DC)
f0 e - S " dx = (1 -1 F(a-I ) for n > 0:
in particular,
. a
2
c --1 ( Ix
Proof. For t := x", we have t" '-1 = 3. 1- " and dt = ox"- Id,.: thus r(Ck
= f •0
C
tdt
.1.
-rr
- 1 di,
-
•0
The last equality is clear, since
r() = \Fr
1),' 2.1.1).
An inductive argument using integration by parts yields x2ne
(Cf. also 1.12.4.6(3).)
= -tr(// + i) 2 2'
n
G
N.
1
2
52
2. The Gamma Function
The Fresnel integrals, already determined in 1.7.1.6*, can also be derived from the l'-integral; for more on this, see Subsection 3. We mention in addition the representation given by F. E. Prym ((Pr], 1876). Partial fraction representation of the F-function. The identity
F(z) =
E z
v=0
v
f 1
0, —1. -2 ...I.
holds for all Z E
Proor The assertion is true for z E T. Since the functions that appear are holothe general case follows from the identity theorem. morphic in C\10. -1, -2, t zi-e dt =- e- '12 F(z), 0 < Re z < 1. To prove this. 3 *. The equation for, let
; then I(C)I < where z = + iy E C, = E C - . Since g E 0(C - ), we have by Cauchy (see Figure 2.1):
g d( =
b) ii I 4
g d(.
If we show that, for 0 < Re z < 1, c)
g d( = lim
11111 •
R
-re
3c_
g d( = 0,
fi
,
the assertion will follow from b) by taking limits, since -y is the path ((t) := it. 6 < t < R. By a),
r
FIGURE 2.1.
• rr/ 2
L
q -
g(renire' o
thus 7r/2
O. For all t > 0, we then have (by (**) of the introduction)
11 1 < (z + 02 - 8 sin2 IS (6 + 0 2 ' Q(t)
-
if z = Izich° with jzi > e and I(P1 < 7r 6. The second integral thus converges locally uniformly in C- by the majorant criterion 3.1. The first integral also converges locally uniformly in C- to the same limit function since -
s P1 (t) Q(t) - 1 z + t dt =
+Jfr S(zQ(t) dt + 02
8
z+t r
for 0 < r < s < oo.
(Integration by parts is permissible because Q is continuous.) We immediately obtain a functional equation for the ix-function:
(3)
(z) - iz(z + 1) = f 2 z
+
t
dt = (z + 1) log(1 +
1) - 1,
z E C-
Proof. Observe that P1 (t + 1) = P1 (t) and write
ii (z + 1) = -
Pi(t + 1)
fo z+t+i
dt =
'
r' I - t
Pi (t)
dt. f z + t dt.= ii(z) .10 z±t 2
+ 1)
The integrand on the right-hand side has (z log(z + tiderivative; (3) follows since log(z + 1) - log z = log(1 +
t) - t
1)
as an anin all of C-.
58
2. The Gamma Function
2. Stirling's formula. For each 6 E (O. j. we denote by
Wh the angular sector fizfr" E Cx < 7r - 61, which omits the negative real axis. The following theorem describes the relationship between the functions F(z) and II(Z:), as well as the growth of tz(z).
Theorem (Stirling's formulas).
(ST)
Proof. Since
l'(z)
=
111 (z)i
5-
vze 1 1 8 cos2 1 1 8 sin 2
z E C- ,
1 IzI 1
z = Ize E z E W6, () < 6 < 7r.
l2 yo > izi sin lb (see the introduc< - s and + IOU 10 this section), the inequalities follow from 1(2). We show, moreover, I hat := E 0(C - ) satisfies the hypotheses of the uniqueness theorem 2.4. The functional equation 1(3) for p(z) immediately gives Q(1)
n z + 1) = (z+ )
= z z+
C
e -z e p (z)
= zF(z).
Furthermore, F is bounded in the strip S E C : 1 < Rez < 21: Certainly e11(z) is bounded there. For all z = x + iy = E C. we have • If z E S and WI 2, then 3' - 0 and jp(iy)1 5_ - 12 Re z
-1 6
for y E R.
The bounds for ii(z) are better than the bounds in 2(1) and 2(2) whenever tan -1(p < icp and cot lb > i(71- - 6), i.e. for (,o < 110.8° and 6 > 69.2°. We will see in §5.3 that p(z) < Alz1-1 in the angular sector 1(p1 < Historical note. C. Gudermann discovered the series A(z) in 1845 [G]; it has sinced been named after him. The inequality with the classical initial factor instead of is due to Stieltjes ([St], p. 443). 5*. Stirling's series. We seek an asymptotic expansion of the function
ti(z) in powers of z -1 . We work with the Bernoulli polynomials Bk(w)=E rc =0
( k )
B
k sc
w k _ _1 k w k-1 +... + Bk, k > 1,
2
62
2.
The Gamma Function
where B„ := B,(0) is the Kth Bernoulli number. The following identities hold (cf. 1.7.5.4): (*) B
1 (w) = (k +1)Bk(iv), k E N.
and
Bk(0) = Bk(1) for k > 2.
To every polynomial B(t), we assign a periodic function P,, : R defined by
(1)
P(t)
R
B„(t) for 0 < t 1.
holomorphic in C- ; furthermore,
(Z) = 1L(Z). 114.(Z) =
B k +1
11(Z) = 1.=1
tk+ (z) . i.t 2 (z) = tt2n+i (z)
k(k +1) zk
;
(3)
1
B2,,
1
(2v - 1)2v
Z2v-1
+ 1t2n+1(z)•
Proof of (3). The recursion formulas follow from integrating by parts in (2): the equations tt2„ = I hold because B3 = B5 = • • • = O. The series in (3) is called Stirling's series with remainder term tt2„ +1 • For n = 1, it(z)
( 3') P3 (1- )
=
1 1 12 z
/33 (t) (z + t) 3
1
where
t3 - -3 t 2 + -i t for t E [0,1].
2
2
IP3 (t)1 < 1/20 (the maximum occurs at 1/2 ± 076) and r z + -2 ( 1.2 +a2)-1/2 t1- 3d1 = is an antiderivative 1 (1z1+ Re 2) -1 (because a s (1 of (x 2 + 2 ) -- 3 / 2 ), it follows that Since
(3
1 60 1z11z1+Rez' 11
")
zEC
Stirling's series (3) does not give a Laurent expansion for p, as n (since it does not have an isolated singularity .at 0). In fact:
For every z E C x , the sequence
B2v
(2v -1)2v
z 2,) --1
oo
is -unbounded.
This follows since 1B2 ,1 > 2(201/(270 2 v (cf. 1.11.3.2) and limn!/rn =oc for 1' > O.
§4. Stirling's Formula and Gudermann's Series
63
The full significance of Stirling's series is realized only through useful estimates for the complicated remainder term. By (3),
1
I-22n -1(Z) = —
r
2n
(4)
P.27(t) n(t) di
B2n
(z + 0 27,
(note that -1 =f k zk 0
dt t)k -E I
(Z
For n > 1, we set M„ suPt>b B2 „ P2n follows immediately that
1 1 1271 -1 1, n 1 2 cos 2n
It 1 2n-i(z)1 < - (2n - 1)2n
(5 )
E IR and z = IzIe L `'4 E C. It
2r
A direct estimate for p271-1(z) without the detour via (4) would have given, instead of (5), only the power izt 2 " -2 in the denominator. (2) and (5) immediately give Z
uni
2n-1
IL2„ - 1(Z) =
B2„ (2n - 1)2n .
From (3) and (5) we obtain the following limit equation for every angular sector W6:
(6)
lirn
z E 14/t,
z-. oc
p,(z) -
E (2v - 1)2v z 2v1 - I B2 v
Izi 2" = 0,
n > 1.
L11
Stirling's series is thus an asymptotic expansion of p(z) (at Dc) -- see 1.9.6.1). If z is large compared to n. this gives a very good approximation for p,(z), but making the index n large for fixed z yields nothing. For n = 3, for example,
1
11 11 11 12 z 360 .-z73 1260 z 5
log F(z) = (z - ) log z - z +log N/-7.Tr+ —
error term.
6*. Delicate estimates for the remainder term. Whoever is ambitious
looks for good numerical values for the bounds M 77 in 5(5). Stieltjes already proved ((St], pp. 434 -436):
(1)
Iti2n-i(z)1
1.
Z=
For n = 1, this is the inequality of Theorem 4. The proof of (1) uses the following unobvious property of the sign of the function P27, (t), t. > 0:
(S)
B277
P2it (t)
always has the sign (-1) 7? -1 ,
71 >
1.
64
2. The Gamma Function
(1) follows quickly from (S): Since B2 7-I (with z = rev') immediately gives COS
2n 1-
( z)I
00
0 arbitrary; dy = O . 3. lini.. f7. I f(s + iy) le-
§4. Stirling's Formula and Gudermann's Series
65
Then 00
f (v) = f (0) + f f (x)dx +
2
i0 -od
f(iu) e 2 Try - 1
dy.
—
0
fo
A proof of this formula can be found in [SG], pp. 438-440. Plana gave his formula (which he described as "remarquable") in 1820 in [P1 ] . Abel arrived at this formula three years later. Cauchy, in 1826, gave one of the first correct proofs; Kronecker treated this and related questions 'in 1889. For further details, see [Lil, pp. 68-69. Clearly the function h(w) := (z + 0 -2 satisfies the hypotheses of Plana's summation formula for every fixed number z E T. We have (cf. Corollary 2.3) 00
h(v) = (log r) u (2), h(0) =
h(r)dx =
1
h(it)-h(-it)
-4izt
2
2 2 = (z + t )
It follows that
(2)
(log r)"(z) =
1
4zt dt (z 2 + t2 ) 2 e 21" — 1 '
+ +f
Re z >0.
Integrating under the integral sign in this equation yields 12(z)
2t
=
z2
O
(3 ) pc(2
)
dt t2 e 2rt
1
2 arctan t/z dt
=
e2/r1
—
, Re z > 0.
1
Proof. Integrating (2) once gives
(o)
1
(log F) / (z) =
+ log z
(it 2t z 2 + t 2 e2 ' - 1
Since
- -1) log
(oo)
log 1"(z) = co + cz + (z - - ) log z -z + f
:= constant.
- 21' = log z - 1/22, integrating again gives 1 2
2 arctan t/z dt. ez,rt _
Comparing (oo) with 4.2(ST') leads to the equation p,(z) =
12 arctan - - log +z+ 1 e 2rt 2 fo'-)c
di.
For the integral 1(z) on the right-hand side, since 0 < arctan t/x < t / x for .r > 0,
0 < 1(x) < -2 xf h ,27rt _ Since
,
thus lin ,„ 1(x) = 0.
it(x) = 0 as well, it follows that co = :12- log 27r and c = 0.
66
2. The Gamma Function
The equations (3) are called Binet 's zntegrals for pi and p. There are other integral representations for p(z); for example,
zr log(1 - e- 2rt , dt 7r j t2
p(z)
Re z > 0.
(Integration by parts gives (3).) All these formulas — except for the Stieltjes formula — are valid only in the right half-plane.
8'. Lindeleif's estimate. A series expansion of 2t/(z 2 + t 2 ) in Binet's integral formula yields formulas for the Bernoulli numbers B2„ and the functions
rx,
t 2n- 1
( _ I ) It - 1
e 2r t _ 1 dt =
io ( _ 1) n-1 = 2 z2 . - 1
(1) 1
dt
z 2 + t2
Jo
e 2-rt _ I '
n -1
t 2v- I
= 2 E( 1)---1 Z 2 ± t2
Z21)
v=1
n> 1 and Re z > 0.
+ (- q)"-1 1 (1 + q), we have (with
(-1)" I qv
Proof. Since 1/(1 + q) :=1.2 /z 2 )
2t
t 2n- 1
foo
n>1
B211)
4n
4 2n -1
- 1 z2
+ 2(
t2
1 z 2n-2
Hence, for Re z > 0, 7(3) gives the series n-1 fi l (Z) = —
(o)
OG
E
0 e 2n t
2v f
v=
dt
t2n-
2(-1)"-' Z2 n - 2
1
—1
0
z2
dt t2 e 2rt
1
On the other hand, differentiating the series in 4.5(3) gives -
(oo)
I
B21., 1 tt (Z)
-
v= 1
2V Z 2v
it2n - 1 (Z)•
For fixed n and large z, the last terms in (o) and (oo) tend to zero like z -2n: In (o) this follows directly; in (oo) one estimates the equation /1 277 -1(Z) :=
B21,P2n (t) (z t)2n4 1
dt
Re z >0.
which comes from differentiating under the integral sign in 5(4). Thus (o) and (oo) are "asymptotic Laurent expansions" for p i (z). The uniqueness of such expansions follows as for power series (cf. 1.9.6.1, p. 294). Comparison of coefficients gives (1).
67
5. The Beta Function
We now estimate p_ i (z). For the function defined in the interval (-;!2 it,
7)
by
e ( o) := 1 for H < ,-1 7T
and
c((,.-..) := I sin 2 y..91 -1 for 1.7r < 14..-1
kir, OW is better than 4.6(1) as long as I sin 2 1 > co5 2 "+ = Interest in delicate estimates like (L) is still alive today. hi their recent article [Sch F] in the venerable Crelle's Journal, W. Schiifke and A. Finstem showed that I sin 2c,.."I -1 , in the angular sector ir < 1*')I < ;17r, is the best of n mdeprndent bounds for which (L) holds. For each individual n, however, there exists a better bound cn(kp) < (2(49) (cf. [Sch
§5.
The Beta Function
The improper integral
(1)
tu
B(tv, z)
'
(1 -
.o
x7= E C2 : converges compactly and absolutely in the quadrant Re z > 0, Re w > 01, and is therefore holomorphic in z E 7 (resp.. u r- 7) for fixed w E T (resp. z E T): the proof, like that for the F-integral, uses a majorant test (cf. also 7.4.2). The function B( w, z) is called the (Euler)
68
2. The Gamma Function
beta function; Legendre referred in 1811 to Euler's integral of the first kind ([L 1 ], vol. 1, p. 221). The main result of the theory of the beta function is
Euler's identity:
B(w,z) =
F(w)r(z) for all w, z E T. r(w + z)
It will be derived in Subsection 1 by means of the uniqueness theorem 2.4.
1. Proof of Euler's identity. We need the following results: a) B(w, 1) =
B(w,z + 1) = ÷u„ . 4,B(w,z),
b) 1B(w, z)1 < B(Rew,Re z). Proof. a) The first formula is trivial; the second is proved as follows:
(w + z)B(w,z + 1) - zB(w,z) pi
= (w + z)
=f
t w-1
(1 - t)z dt - z] tw -1 (1 - tr l dt
13
1
{wtw -1 (1 - t)z - tw z(1 - tr i ldt
= [tw(1 - t)1(1) . O. b) This is clear, since 1(1 - t)w -i tz -1 1 < ( 1 _ oRe w—ltRe z — 1 To prove Euler's identity, we now fix w E T and set F(z) := B(w, z)F(w+ z) E 0(T). By a), F(1) = r(w) and F(z +1) = zF(z). Since 1r(w + z)1 < F(Re(w + z)), b) shows that F is bounded in the strip {1 < Re z < 2 } . By the uniqueness theorem 2.4, F(z) = F(w)F(z). A proof of Euler's identity for real arguments, using the logarithmic convexity of the product B(x, y)r(x + y), can be found in Artin's book ([A], pp. 18-19).
Because of the formula B(w, z) = F(w)F(z)/r(w + z), the beta function is not interesting in its own right. Despite this, it survived for quite a while alongside the gamma function as a separate function: a profusion of relations between beta functions was derived, especially by means of the identities a); these often reduce to trivialities as soon as the Euler identities are applied. See, for example, the classical works of Legendre ([L 1 ,4 passim) and Binet [Bin], or even those of Euler himself (and also [Ni], p. 15). The following integral formulas, valid for all w, z E T, are useful: 1„
(1)
B(w, z) = 2 f0
2
OC
s
w-i
(sin '' (p) 2w-1 (cos cp) 2.z-14 = fo (1 + s)w-i-z ds.
Proof. In (1) of the introduction, substitute t = sin2 (p (resp., s = tan 2 (p); thus (1 + cos2 go and d.s = 2 tan (p(cos )24.
§5. The Beta Function
69
Historical note. Euler, in 1766, systematically studied the integral IV -
(*)
Jo xP-1 (1 —
'
= fo
([Eu], I-17, pp. 268-287); he writes () for his integral. Substituting y := xn yields
(E) n
n
)
n)
Integrals of the type (*) already occur in Euler's "De productis ex infinitis factoribus ortis," which was submitted to the Petersburg Academy on 12 January 1739 but not published until 1750 ([Eu], I-14, pp. 260-290). Euler knew by 1771 at the latest that the beta function could be reduced to the gamma function (cf. [Eu], 1-17, p. 355). of Euler's identity. Because B is holomorphic in 7, it suffices to verify the formula for real numbers w > 0, z > 0 (identity theorem) 2. Classical proofs
Dirichlet's proof (1839, [D], p. 398). First, we have Do
(1 + s )- zr ( z ) = f tz- 1 e - (1+ s)t dt.
Re s > —1. z > O (even z E T). 0
Substituting w + z for
z
and using 1(1), we find that
F(w + z)B(w,z) = 00 s w-1
[130 tw
+ z -1 e
-(1 +s )f dt (Is, w > 0, z > 0.
By theorems of real analysis, reversing the order of integration is legitimate here for all real w > 0, z > 0 (!); thus F(w + z)B(w,z) = The inner integral equals
0f 0
F(w)t'.
, e -ts ds t w+z-i c -e fit. .s 1,--t
Hence
F(w + z)B(w,z) = F(w) f
= r(w)F(z).
Dirichlet carefully examined the theorem used to reverse the order of integration. Jacobi, in 1833, argued concisely as follows [J]:
70
2. The Gamma Function
Demonstratio formulae ,
w)b-i a w =
f
a x.r e—xx1-4 ax •
e-xx'344-40x
—
ra rb ro +6) 9
(Auct. Dr. C. G. J. Jacobi, prof. mad,. negiont.) uoties variabilibus x, y valores omnes positivi tribuuntur Q quo ad +00, posit°
hide a 0 us-
x+y=r, x=rw, variabili novae z valores conveniunt mimes positivi a 0 usque ad +00, variabili w valores omnes positivi a 0 usque ad +1. Fit simul axay = raraw. Sit iam o notatione nota: r (a) =f e-xxa-' a x, babotur r (a) (6)
r
variabilibus x, y tributis valoribus omnibus positivis a 0 usque ad + co. Posit° autem: x+ y = r, x = r w, integrate duplex propositum ex antecedentibus alter° quoque modo in duos factores discerpitur:
r (a) r (6) = ino ce Ce—r 143+6 -1 a r.1 1 wa-1 (1— w)b-i a w,
untie fi to-1(1— wy-1 a w
r (n)
r(b)
(a+ 1,) •
Quod est theorema fundamentale, quo integralium Eulerianorum, quae ill. Legendre vocavit, altera species per alteram exhibetur. 23. Aug. 1833. Exercises. Prove the following identities. P(n ---,l,P for m, n E NM01. 1) fo' 12 (cos (p) 2 m -1 (sin (p) 2 ' 1 4, = 2 ( 77,-1 (M+ 71- r
2) Hin"irz =
dt = 2 fo' (tan
‘19
) 22-1 cbp =
1.00
s z-1 ds for 0< Be z < 1. o i+s
Bibliography [A]
ARTIN, E.: The Gamma Function, trans. M. BUTLER, Holt, Rinehart and
Winston, New York, 1964.
[Bi]
BIEBERBACH, L.: Theome der gewdhnlzchen Differentialgleichungen, Grdl.
Math. Wiss. 66, 2nd ed., Springer, 1965. [B]
BINET. M. J.: Mémoire sur les intégrales définies Eulériennes, Journ. de
l'École Roy. Polyt. 16, 123-343 (1839). [BN,11
Bonn, H. and J. MOLLERUP: Laerebog z matematisk Analyse, vol. 3. Copenhagen, 1922.
71
§5. The Beta Function
DIRICHLET. P. G.: Über eine neue Methode zur Bestimmung vielfacher Integrale, Abh. Ke•nigl. Preuss. Akad. Wiss. 1839, 61 - 79; Werke 1, 393 -
[D]
410. [Eul
Leonhardi EULERI Opera omnia, sub auspiciis societatis scientarium naturalium Helveticae, Series I-IV A, 1911-.
[G 1
GAUSS,
]
[ G2)
C. F.: Letter to Bessel, Werke 10, Part 1, 362-365.
GAUSS, C. F.:
a •
1•
x+
Disquisitiones generales circa seriam infinitam
a(a + 1)43 + 1) (a + 1)(a + 2)0(0 + 1) (3 + 2) 3 x + etc., xx+ 1+ 1 • 2 • y ( y + 1) 1 • 2 • 3 • y ( y + 1)(-y + 2) -
-
-
-
Werke 3, 123-162. xa-
Gui
Additamentum ad functionis F(a) = f, theoriam, Journ. reine angew. Math. 29, 209-212 (1845).
[H]
HANKEL, H.: Die Eulerschen Integrale hei unbeschrnkter Variabilitk des Argumentes, Zeitschr. Math. Phys. 9, 1-21 (1864). HAUSDORFF, F.: Zum H61derschen Satz über F(x), Math. Ann. 94, 244-247 (1925). WILDER. O.: Ueber die Eigenschaft der Gammafunktion keiner algebraischen Differentialgleichungen zu genügen, Math. Ann. 28, 1-13 (1887).
[Ha] [ 116]
GUDERMANN, C.:
.1Ac0m, C. G. J.: Demonstratio formulae fol wa -1 (1 - w) b- I dtv = Journ. reine angew. Math. 11, p. 307 (1833); Ges. Werke 6, 62 - 63. [Je]
JENSEN: An elementary exposition of the theory of the gamma function, Ann. Math. 17 (2nd ser.), 1915-16, 124-166.
[Kn.]
KNESER, H.: Funktionentheorze, Vandenhoeck & Ruprecht 1958.
[Ku]
KUMMER, E. E.: Beitrag zur Theorie der Function r(x) = f(;.' e Journ. reine angew. Math. 35, 1 - 4 (1847): Coll. Papers II, 325 - 328.
[LL]
LEGENDRE. A. M.: Exercices de calcul intégral, 3 vols., Paris 1811, 1816, and 1817.
[L2]
LEGENDRE:, A. M.: Traité des fonctions elliptiques et des intégrales Eulér-
in
-
.
dr
-
iennes, 3 vols., Paris 1825, 1826, and 1828. [Li]
LINDEL6F, E.: Le calcul des résidus, Paris 1905; reprinted 1947 by Chelsea Pub!. Co., New York; Chap. IV in particular.
[MI
MOORE. E. H.: Concerning transcendentally transcendental functions, Math.
Ann. 48, 49-74 (1897).
[Ne]
NEWMAN. F. W.: On
[Ni]
NIELSEN, N.: Handbuch der Theorte der Gammafunktion, first printing Leipzig 1906, reprinted 1965 by Chelsea Pub!. Co., New York.
[ 0]
OSTROWSKI, A.: Zum H61derschen Satz über F(x), Math. Ann. 94, 248-251 (1925); Coll. Math. Pap. 4, 29 32. P1NCHERLE. S.: Sulla funzioni ipergeometriche generalizzate, II, Atti. Rend. Reale Accad. Lincei (4). 792 - 799 (1889), Opere scelte I, 231 - 238.
[Pi]
F(a) especially when a is negative, Cambridge and Dublin Math. Journ. 3, 57 - 60 (1848).
72
2. The Gamma Function
PI]
PLANA, J.: Note sur une nouvelle expression analytique des nombres Bernot liens, propre à exprimer en termes finis la formule générale pour la sommation des suites, Mem. Reale Accad. S'cz. Torino 25, 403-418 (1820).
. Pr]
PRYNI, F. E.: Zur Theorie der Gammafunction. Journ, reine angew. Math. 82, 165 172 (1876).
REMNIERT, R..: Wielandt's theorem about the F-function, Amer. Math. Monthly 103, 214-220 (1996).
RENIMERT, R.: Wielandt's characterization of the ['-function. in Math.
Works of Helmut Wzelandt, vol. 2, 265-268 (1996). "Sala]
SCHARLAI'. W. (ed.): Rudolph Lipschitz, Briefwechsel mit Cantor, Dedekin( Helmholtz, Kronecker, Wezerstrass und anderen, Dok. Gesch. Math. 2, DMV, Vieweg u. Sohn, 1986.
[ Sal
SCH .AFKE, F. W. and A. FINSTERER: On Lindela's error bound for Stirling's series, Journ, reine angew. Math. 404, 135-139 (1990).
[Sch S]
SCICAFKE, F. W. and A. SATTLER: Restgliedabschàtzungen filr die Stirlingsche Reihe, Note. Mat. X, Suppl. no. 2, 453 470 (1990).
[Sc hi]
SCHL6MILCH, O.: Einiges liber die Eulerischen Integrale der zweiten Art, Arch. Math. Phys. 4, 167-174 (1843).
[SG]
SANSONE, G. and J. GERRETSEN: Lectures on the Theory of Functions o, a Complex Variable, vol. I, P. Noordhoff-Groningen, 1960.
(St)
STIELTJES, T.-J.: Sur le développement de log F(a), Joui-n. Math. Pur.
Appl. (4)5, 425-444 (1889); (Bums 2, Noordhoff. 1918, 211-230; 2nd ed.. Springer, 1993, 215 234. [T]
TiTcHmARsti. E. C.: The Theory of the Riemann Zeta-Function, Oxford University Press. 1951.
[W]
WALTER, W.: Analysis I, Grundwissen 3. Springer, 1985, 2nd ed. 1990.
[Wei]
WEIERsTRAss, K.: Über die Theorie der analytischen Facultaten, Journ reine angew. Math. 51, 1-60 (1856), Math. Werke 1. 153-221.
[We2]
WEIERSTRASS, K.: Zur Theorie der eindeutigen analytischen Functionen Math. Werke 2, 77-124.
[WW]
WHITTAKER. E. T. and G. N. WATSON: A Course of Modern Analysis 4th ed., Cambridge Univ. Press, 1927, Chap. XII in particular.
3 Entire Functions with Prescribed Zeros
Es ist also stets miiglich, eine ganze eindeutige Function G(x) mit vorgeschriebenen Null-Stellen (11, az, a3. zu bilden, wofern nur die nothwendige Bedingung LimIa n I = oc erffillt ist. (It is therefore always possible to construct a single-valued entire function G(x) with prescribed zeros al, az, a3. • • •, provided only that the necessary condition Ian' = oc is satisfied.) — Weierstrass, Math. Werke 2, p. 97
If f 0 is a holomorphic function on a domain G, its zero set Z(f) is locally finite in G by the identity theorem (cf. 1.8.1.3). It is natural to pose the following problem: Let T be any locally finite subset of G, and let every point d E T be assigned a natural number D(d) > I in some way. Construct functions holomorphic in G which each have zero set T and, moreover, whose zeros at each point d E T have order (d).
It is not at all clear that such functions exist. Of course, if T is finite, the polynomials
fl
dET
(z _ (owl)
Or
2 (13)
H dET\{o}
z)D(d) 1— — d
give the desired result (the initial factor z' (()) appears only if 0 E T). In 1876, Weierstrass extended this product construction to transcendental
74
3. Entire Functions with Prescribed Zeros
entire functions: for a prescribed sequence d, E C x with lim d, = oc, he constructs products of the form ,m n
v>i
[
d,
1 I z\ 2 1 / z \ d, + 2- -d, -- ) ± • " ± Z T ,)
and forces their normal convergence in C by an appropriate choice of natural numbers k,. The novelty of this construction is the use of nonvanishing conveigence-producing factors (for historical details, see 1.6). We study Weierstrass's construction in detail in Section 1 and discuss its applications in Section 2.
§1.
The Weierstrass Product Theorem for C
The goal of this section is the proof of the Weierstrass product theorem for the plane. To formulate it properly, we make use of the concept of divisors. In Subsection 1, with h a view toward later generalizations, we define divisors for arbitrary regions D in C. Theorem 2 describes the simple principle by which Weierstrass products are used to obtain holomorphic functions with prescribed divisors. In order to apply this theorem, we introduce the Weierstrass factors En (z) in Subsection 3. They are used in Subsection 4 to construct the classical Weierstrass products for the case D = C. Subsection 5 contains elementary but important consequences of the product theorem.
1. Divisors and principal divisors. A map : D Z whose support S := E D : D 01 is locally finite in D is called a divisor on D. Every function h. meromorphic in D whose zero set Z(h) and pole set P(h) are discrete in D determines, by z o(h), a divisor (h) on D with support Z(h) U P(h); such divisors are called principal divisors on D. The problem posed in the introduction to this chapter is now contained in the following problem:
Prove that every divisor is a principal divisor. We begin by making a few general observations. Divisors D, (as maps into Z) can be added in a natural way; the sum D + D is again a divisor (why?). It follows easily:
The set Div(D) of all the divisors on D is an abelzan group, with addition as group operation. A divisor D is called positive and written D > 0 if D(z) > 0 for all z E D; for obvious reasons, positive divisors are also called distributions of zeros. Holomorphic functions f have positive divisors (f). The set ./vt (D )x of all fund ions meromorphic in D that have discrete zero sets is a multiplicative abellan group; more precisely, .A4(D) x is the group of units of the ring
§1. The Weierstrass Product Theorem for C
M(D). If D = G is a domain, then M(G) is a field; thus ../v1(G)x »f(G)\{0} (cf. 1.10.3.3). The following is immediate. The map .A.4(D)x Div(D), (h), is a group homornorphasm. Moreover, 1) f E M(D) x is holoindrph,ic in D 0; 2) f E M(D) x is a unit in 0(D) 1 is called a Weierstrass product for the divisor D > 0 in D if the following conditions hold. 1) fi, has no zeros in D\{cl,} and o(/,(f,) = 1, u> 1.
2) The product 11, >1 f, converges normally ri D. This terminology will turn out to be especially convenient; the next result is immediate. Proposition. If f is a Weierstrass product for D > 0, then (f) = 0: that is, the zero set of f E 0(D) is the support T of D, and every point d E T is a zero of f of order 0(d).
3. Entire Functions with Prescribed Zeros
76
Proof. By 2), f E 0(D). Every point d E T. d 0, occurs exactly 0(d) times in the sequence d,,; hence 1) and Theorem 1.2.2 (applied to the connected components of D) imply that oz (f) = D(z) for all z E D. Therefore
(f) = The next statement follows immediately from the definition. If zD" fl f,, and z5(0) f,, are Weierstrass products for D > 0 and > O, respectzvely, then zp(0)+5(0) rig is a Weierstrass product for D -6, where ,
g2v - := Li and g2v := We will construct 'Weierstrass products for every positive divisor D. (This involves more than finding functions f E 0(D) with (f) = D.) In the construction, the "only" thing that matters is choosing the factors f,, E 0(D) in such a way that 1) and 2) hold. When D = C, such factors can be specified explicitly.
3. Weierstrass factors. The entire functions
E0(Z) := - Z,
Ei ,(Z) := (1
-
Z2 Z3 Z) exp (z + — + — + • • • +
2
n > 1,
3
are called Wezerstrass factors. We observe immediately that
z"
z2 (1) En1 (z) -z" exp (z + -y + • (2) E„(z) = 1 +
n for
E ay e, where E
v>,1
n > 1;
a,, = 1, for n > O.
v>n
Proof. Let t„(z) := z + z 2 /2 + • + z" /n; then (1 - z)t ra (z) = 1 - zn. ad (1): Write K(z) = - expt„(z)+(1-z)t in (z)expt,(z) = -znexpt„(z). ad (2): Let E a,,zu be the Taylor series for E„ about O. The case n-= 0 is trivial. For n > 1, we have E vav zv -1 = -zn expt„(z) by (1). Since the function on the right-hand side has an nth-order zero at 0 and all the Taylor coefficients of exp t„(z) about 0 are positive, we see that
= • • • = a„ = 0 and o„ < 0;
thus a,, = -a,, for u > n.
(2) follows because a() = E„(0) = 1 and 0 = E„(1) = 1 +
E
ay.
From (2), we immediately obtain
(3) En(z) -
14' 1 . n= 0, 1, 2,.... for. all z E C withlzi r. Since lim141 = oo, for every r > 0 there exists an n(r) such that for y > n(r). Hence
E
v>n(r)
lEku (z/d,) _
< E
< oc for every
>r
7' >
v>n(r)
proving the normal convergence of the product. Since the factor Ek.,,(Z/c1„) E 0(C) has no zeros in C\ {du } and has a first-order zero at (1, we have a Weierstrass product for D. Product theorem. For every divisor D > 0 on C. there exist Weierstrass products, e.g.
78
3. Entire Functions with Prescribed Zeros
z" H Ei,_ 1 (zId„) v>1 =
2(0) H
exp (-jz— 4.
12.
2v 1 Gfd y 1 )1 . -1
v>1
Proof. Given r > 0, choose m E N such that Id y l > 2r for v > m. It follows that < Eu>ni 2' < oc. Thus (1) holds for k„ := u — 1. The choice k, := y — 1 is not optimal. It suffices, for example, just to require that k,, > a log v with a > 1: since Idy l > e • r for all but finitely many v, we have Ir/d„i k v 41 < v, so that (1) holds.
5. Consequences. The product theorem 4 has important corollaries.
Existence theorem. Every divisor on C is a principal divisor. Factorization theorem. Every entire function f the form f (z) = eg ( z ) z'n
H
[( 1
exp ( z z
0 can be written in
(zz ) 2 ±
(z
k
)]
)
v>1
E
where g 0(C) and and zni product for the divisor (f).
11 >1 _
...
is a (possibly empty)
Weierstrass
Only the factorization theorem needs justification. By the product theorem, there exists a Weierstrass product f for the divisor (f). Then f/f is a function without zeros, and thus of the form expg with g (cf. 1.9.3.2).
E 0(C)
The next result. is a simple consequence of the existence theorem. Theorem (Quotient representation of meromorphic functions). For every function h meromorphic in C, there exist two entire functions f and g, without common zeros in C, such that h = f g. Proof. Let h O. Positive divisors on C with disjoint supports are defined by b+ (z) := max{0, o z (h)} and D+ (z) := max{0, —o(h)}; they satisfy O. For 0(C) be chosen with (g) = V. Then g (h) = D+ — V. Let. g f := gh, it follows that (f) = (g) + (h) = > 0, whence f is holomorphic in C. By construction, Z(f) n Z(g) is empty.
E
In particular, we have proved the following: The field M(C) of functions meromorphic in C is the quotient field of the integral domain 0(C) of functions holomorphic in C.
§1. The Weierstrass Product Theorem for C
79
The theorem contains more than this last statement: for an arbitrary quotient
f Ig, the numerator and denominator may have infinitely many common zeros; without the existence theorem, it is not clear that these zeros all cancel out.
We conclude by noting a Root criterion. The following statements about an entire function f 0 and a natural number n > 1 are equivalent: i) There exists a holomorphic nth root of f; that is, there exists a g E 0(C) with gn = f. ii) Every natural number o(f), Z E C, is divisible by n. Proof. Only the implication ii) i) must be proved. By hypothesis, there exists a positive divisor D on C with riD = ( f). Let Ti E 0(C) be chosen such that CO = D. Then := is holomorphic and nonvanishing in C; hence there exists u G 0(C) with = un (existence theorem for holomorphic roots; cf. 1.9.3.3). The function g := u is an nth root of f.
The existence theorem allows us to prescribe the location and order of the poles of rneromorphic functions. We will see in Chapter 6 that, in doing so, we can also arbitrarily prescribe all principal parts. But the following is immediate from the product theorem 4, by logarithmic differentiation of Weierstrass products.
(1) Let 0, d 1 , d2 , . . . be a sequence of pairwise distinct points in C that have no accumulation point in C. Then the function
1
. +E(
1
1
±
is meromorphic in C and holomorphic in CVO, d 1 , d2, .. .1 ; it has principal v > 1. part (z - 4) -1 at
6. On the history of the product theorem. Weierstrass developed his theory in 1876 (M i l. pp. 77-124). His main objective was to establish the "general expression" for all functions meromorphic in C except at finitely many points. "[Da.zul hatte ich jedoch zuvor eine in der Theorie der transcendenten ganzen Functionen bestehende Liicke auszuffillen, was mir erst nach manchen vergeblichen Versuchen vor nicht langer Zeit in befriedigender Weise gelungen ist." (To do this, however, I ... first needed to fill in a gap ... in the theory of transcendental entire functions, which, after a number of futile attempts, I succeeded only recently in doing in a satisfactory way.) ([W 1 ], p. 85) The gap he mentioned was closed by the product theorem ([W 1 ], pp. 92-97). What was new and, for his contemporaries, sensational in Weierstrass's construction was the application of convergence-producing factors that have no influence on the behavior of
80
3. Entire Functions with Prescribed Zeros
the zeros. Incidentally, according to Weierstrass ([Wi I, p. 91), the idea of forcing convergence by adjoining exponential factors carne to him by way of the product formula
1,r(z).z ri {(1,)(v+i) v
u?'
z}.
z
{(i +
v>1
'l-
z i'g(} ,
which lie attributes to Gauss rather than Euler; cf. 2.2.2. In 1898 H. Poincaré, in his obituary for Weierstrass, assessed the discovery of the factors E7 (z) as follows OP21, p. 8): "La principale contribution de Weierstrafi aux progrès de la théorie des fonctions est la découverte des facteurs primaires." (Weierstra13's major contribution to the development of function theory is the discovery of primary factors.) Special cases of the product theorem had already appeared in the literature before 1876, for example in the work of E. Betti (cf. 2.1). The awareness that there exist entire functions with ''arbitrarily" prescribed zeros revolutionized the thinking of function theorists. Suddenly one could "construct" holomorphic functions that were not even hinted at in the classical arsenal. Of course, this freedom does not contradict the solidarity of value behavior of holomorphic functions required by the identity theorem: the "analytic cement" turns out to be pliable enough to globally bind locally prescribed data in an analytic way. From his product theorem. Weierstrass immediately deduced the theorem on quotient representation of meromorphic functions ([W 1 ], p. 102). He attracted attention by this alone. No less a figure than H. Poincaré seized this observation of the "célèbre géomètre de Berlin" arid carried it over to meromorphic functions of two variables [P1]. With his theorem on the representability of every function meromorphic in C2 as the quotient f (w, z) I g(w, z) of two entire functions in C2 (locally relatively prime everywhere), Poincaré initiated a theory that, through the work of P. Cousin, H. Cartan, K. Oka, J-P. Serre, and H. Grauert, is still alive today; see the glimpses in 4.2.5, 5.2.6, arid 6.2.5.
§2.
Discussion of the Product Theorem
When we apply the product lemma 1.4, we will choose the numbers is:, as small as possible, in accordance with the idea that the smaller k, is, the simpler the factor Ek y (z I d„). Situations in which all the k, can be chosen to be equal are especially nice; they lead to the concept of the canonical product (Subsection 1). In Subsection 2 we show that not only the Euler products of Chapter 1.4 but also the sine product and the product H(z). so important for the theory of the gamma function, are canonical Weiers.trass products.
§2. Discussion of the Product Theorem
81
In Subsections 3 and 4 we discuss the a-product and the p-function. We prove that a(z; wi, w2 ) and K)(z; w , w2) are holomorphic and meromorphic, relpectively, in all three variables. Since the time of Eisenstein and Weierstrass, these functions have been central to the theory of elliptic functions. Subsection 5 contains an amusing observation of Hurwitz. 1. Canonical products. Let D again denote a positive divisor on C and d1 , d2, ... a corresponding sequence. We first make a few observations. (1) If f (z) = F1(1– z d,)ePv ( z ) converges normally in C and every function p„ is a polynomial of degree < k, then E 11/411-4-1 converges. Proof. Differentiating f' (z)I f(z) = E [1/(z - d,) + p',(z)1 k times yields the series E(-1)kk!/(z - d,,)k-i-1. which converges absolutely at 0 E C. 0 We now ask when, for a given D, there exist Weierstrass products of the particularly simple form zD ( ° ) n ,,,, Ek ( z / do with fixed k G N.
(2) 2 0) 11, >1 Ek(z1d,) is a Weierstrass product for the divisor D if and only if E 11/44 k -f-1 EA. ( z / d, ) is the canonical product for D if and only if
E
oc
and
E
0 is such thatldii –d,i > m for all ki 1/, then E it/ci v i" < oc for a > 2. In this case, there exists a canonical product for D with k < 2.
82
3
Entire Functions with Prescribed Zeros
ithstorical note. E. Bet ti. in 1859 - 60, proved (3) in order to write elliptic functions as quotients of theta series; cf. the article by P. Ullrich ([Ul, p. 166).
2. Three classical canonical products. 1) The product
H
+
)=
ET> Eg-gu z), v
1, -> 1
where
0 < < 1,
I
discussed in 1.4.3, is the canonical product for the divisor on C given by
:= 1 for u = 1, 2, ... ;
:= 0 otherwise.
(Pu)position 1 holds with k := 0.) 2) 'The function -
=--
"/F(z) = z H(1+ v v>1
E 1 (--). vi
considered in 2.1.1. is the canonical product for the divisor on C defined by 0(-v) := 1 for v E N, 0(z) := 0 otherwise.
(Proposit ion 1 holds with k 3) Tlie sine product
1 but not with k := 0.)
,2 -
7)
V
'
z
11 1( 1 -
)c." 11
(
+
v>1
zH v>1
is the canonical product for the divisor on C defined by
D(v) :=-- 1 for v E Z;
D(z) = 0 otherwise.
(Proposition 1 holds with k := 1 but not with k := 0; a corresponding sequence is 1, --1, 2, -2,
In lectures and textbooks, these examples are sometimes given a.s examples of applications of the \Wierstrass product theorem. This is misleading. These products were known long before Weierstrass. Of course, his theorem shows that the sanie construction principle underlies them all. ercr.scs. Determine the canonical product. for of the following s (' quences: a) (ji , •---= ( - I)
b) d,, :=
ply,
> 0 in C corresponding to each
V > 1;
where p E N with 4// - 3 < v 1
§2. Discussion of the Product Theorem
83
3. The a-function. If w1, w2 E C are linearly independent over R, the set Q := 7 + Zw2 = = mw i + nw2 : m, n E Z1 is called a lattice in C. Q is locally finite in C and
:C
if z E Q, if z Q,
6(z) := { 01
N, • z
is a positive divisor on C with support Q. Proposition. The entire function a(Z,
(1) a(Z)
H
Q) := z
/
(i
00wES-2
-
\2
=z
z
(.4)
E2
is the canonical Weierstrass product for the lattice divisor 6. The proposition is contained in Bettis result 1(3). We give a direct proof, which even yields the normal convergence of the a-product (1) in all three variables z, w 1 , and w2. The set U := {(2i,v) E C2 : u/v E HI is a domain in C2 . For every point (w i ,w2) E U, the set Q(w i , w2 ) := Zwi + 2 is a lattice in C; conversely, every lattice Q C C has a basis in U. The following lemma is now crucial. Convergence lemma. Let K C U be compact and let a > 2. Then there exists a bound M > 0 such that
E
< M for all (w1, w2) E K;
PI
0#wEs2(wi.w2)
E 00u.; ES 2(.4.)
w1
-2
=
°
DI '
)
Proof. The function
q : (R2 \{(0,0)} x U
(a:,
, w2)
ixc.o
yw2 1/
1.2 ± y2
is homogeneous in x y; hence q(R2 \{(0,0)} x U) = q(S1 X U). Since q is continuous, it has a maximum T and a minimum t on the compact set 5 1 X K. The R-linear independence of w 1 , w2 implies that q is always positive; hence t > O. Since ,
tV771. 2 + n2 < I m‘.4) 1 + nw2 0, > 0, (10 > d. Such series (with 13 = 2) were considered by Eisenstein in 1847 ( Werke, pp. 361-363). A variant of the proof was given in 1958 by H. Kneser ([Kri], pp. 201 202). He replaces q by the function Ixwi +yw2I/max(141y1). As above, there exist numbers S > s > 0 such that s < Irnw i + nt.u21/ max(Iml, Ind) < S. The convergence of E lwl - is now equivalent, to that of 0,
[max(Iml.ini)r =
00
E
+
rn=1
0,(rn,n)EZ2
E
[max(m,n)]°.
rn,n=1
But the series on the right-hand side can be written as follows (!):
E n + E(k - 1)k = E(27/'-a
E
(nn +
n=1
rn=n-4-1
k=I
n=1
this converges for a > 2 and diverges for a = 2.
'For -y > 0, noting that
E
n 1+-1
1
v!
-•, z I q2.z + - -
.
Expanding the right-hand side in powers of z gives Taylor coefficients a,, that indeed lie in Q(i), since each a,, is a polynomial with rational coefficients in finitely many of the ch. q), . E Q( t). If the power series of f about 0 has only real coefficients. then all the b,, with v > 1 are real. In this case. one can always choose gt, E Q and hence a,, C Q. At this point, Hui wit z's observation is quickly proved. We choose f E 0(C) with (f) = D. Then D is also the divisor of every function g := f expg. g 0(C). By the lemma, g can be chosen in such a way that. all the Taylor coefficients a,, of g belong to Q(i).
D(z), then D is also the divisor of the entire function j whose Taylor coefficients are the numbers cit,. Then 20 is the divisor of g g- -: by the root criterion, there exists c 0(C) with 42 = q. Moreover, (4) = D. Since all the Taylor coefficients of gej are rational real numbers and the first nonzero coefficient is positive, all the Taylor coefficients of are rational real numbers. ID If it always holds that DM
Hurwitz proved the preceding assertion in 1889. As an amusing corollary, he also noted the following: Every (real or complex) number a (thus. for instance, f: or 7r) is the root of an equation 0 = 1 .0 + 7. 1 Z whose right-hand side is an entire function 1 . 2E 2 with rational coefficients (real or complex. respectively). which has no roots other than a. + • • •
Bibliography FEJ(it, L : ( .Tber die Weierstrassche Primfunktion. Ces. ,4rb. 2, 849-850.
(IN
Htt,t.t.„ E.• Analytic- Function Theory, 2 :vols., Ginn and Company. 1959 and 1962. litinwrrz. A.: Ube! bestLindig convergierende Potenzreihen mit rationalen Zahlencoeffizienten und Nrorgeschriebenen Nullstellen, Acta Math. 14, 211215 (1890 91); Matti. Werke 1, 310-313.
§2. Discussion of the Product Them em
87
KNESER. H.: Funktionentheorie. Vaudou hoeck X, H uprecht 1958 ORLAND°. L.: Sullo sviluppo della funzione (1 - z)c z 1 ' 21 nale Matent. Battaglini 41. 377-378 (1903).
[Pd
POINCARÉ. H.. Sur les fonctions do deux variables, Acta Math
, Cior2, 97 113
(1883); Œuvres 4. 147-1,61. il) 21
POINCARÉ. H.: L'oeuvre math6matique de Weierstrali, Acta Math. 22, 1 18 (1898); not in Poincaré's (Euvïcs ULLRICH. P.: Weierstraii . Vorlesung zur "Einleitung in die Theorie der analytischen Funktionen," Arch. Hist. Ex. Sc. 40, 143 172 (1989)
[W d
WEIERSTRASS, K.: Zur Theorie der eindeutigen analytischen Fumet ionen, Math. Werke 2, 77-124.
Y2]
WEIERSTRASS, K.: Vorlesungen iiber die Theorie der elliptischen Funktionen. adapted by J. KNOBLAUCH, Math. Werke 5.
4* Holomorphic Functions with Prescribed Zeros
We extend the results obtained in Chapter 3 for entire functions to functions holomorphic in arbitrary regions D in C. Our goal is to prove that every divisor on D is a principal divisor (existence theorem 1.5). For this purpose we first construct, in Section 1, Weierstrass products for every positive divisor. As before, they are built up from Weierstrass factors En. and converge normally in regions that contain C\OD (product theorem 1.3). In Section 2 we develop, among other things, the theory of the greatest common divisor for integral domains 0(G). Blaschke products are a special class of Weierstrass products in E; they are studied in Section 3 and serve in the construction of bounded functions in 0(E) for prescribed positive divisors. In an appendix to Section 3 we prove Jensen's formula.
§1.
The Product Theorem for Arbitrary Regions
A convergence lemma is proved in Subsection 1. In Subsection 2, Weierstrass products are constructed for some special divisors; the factors En (z 1 d) are now replaced by factors of the form d
( 61— \Z — CJ
\Z
—
)
CJ
exp
[d — c 1 (d — c)d 2 2 z—
±1
(
— c) 71 — C)
1
c d, which also vanish to first order at the point d. The general product theorem is derived in Subsection 3.
90
4*. liolomorphic Functions with Prescribed Zeros
1. Convergence lemma. Let D be a positive divisor on D with support T. From the points of the countable set T, we somehow construct a sequence (d„) in which every point d E T occurs exactly 0(d) times. (In contrast to our earlier discussion in 3.1.2 -- the origin, if it lies in T, is not excluded from the sequence.) The following is a substitute for Lemma 3.1.4. Lemma. Let (e,),>, be a sequence in C\D and (k„),>1 a sequence of
natural numbers such that
E Ir(d„ - c)l"
O.
3C
v=i Then the product (du - c 1t ) - c„
H z de ity )
p >1
+ +
1 (d, z
[(dt, et/ ) exp z - e i, kv ]
1 ( )
2
eV
2
z - c,
et/
converges normally in C\{ei, ca....} D D; it is a Weierstrass product in D for the divisor D. Proof. We set S :-= {e 1 ,c21 ...} . For f(z) := Ek p Rdi) — Cv V(Z have (*)
0 if z
fu E 0(C\S), f(z)
du ,
and
— co)],
we
od,(M =1.
Let K be a compact set in the region C\S. For all z E K, lz - c u l > d(K , c„) d(K S) > 0; hence i(d„ - c„)/(z - c v )IK < rid, - cv l, where r := d(K , S) - '. Since urn id„ - c„1 = 0 by (1), there exists n(K) G N such that rid, - ev i < 1 for v > n(K). Since 1E 1 (w) - 1 < iwin+ 1 for w E E by 3.1.3(3), it follows that
E
f
ir(d, - c„)ik ,4-1
- 111%:
v>r?(K)
This proves the normal convergence of is a Weierstrass product for D in D. Corollary to the lemma. If
product
nu>, Ed ( d,, _
Proof. (1) holds with k„ := k.
co.
v>71(K)
_
E
ri f, in C\S. By (*), this product
d, - c1.,' < co for some k E N, the eu )] is a Weierstrass product for D in D.
§1. The Product Theorem for Arbitrary Regions
91
2. The product theorem for special divisors. In general, the region of convergence of the product constructed in Lemma 1 is larger than D. As the zero set of the product, T is closed in this larger region. We make a general observation, leaving the proof to the reader:
(1) U T is a discrete set in C. then the set T' := T\T of all the accumulation points' of T in C is closed Z7/ C. The region C\T' is the largest subset of C in which T is closed. By (I), every positive divisor D on D with support T can be viewed as for a positive divisor on C\T' D D with the same support (set D(z) := z E (C\T)\D). Clearly r D OD. The next theorem now follows quickly from Lemma 1. Product theorem. Let D be a positive divisor on D with corivsponding sequence (d,,),>1. Let a sequence (c,,)„>1 in T' be given such that him il,, — E„_ i Rd„ — c„)/(z — e,,)1 is a, Weierstrass c„1 = O. Then the product product for e n C\T'
ri
E
—
cor < cx, for every Proof Since lim d,, — e , = 0, it follows that r > O. Hence 1(1) is satisfied with k,, := v-1. Now we have fc 1 , c2, C (in fact, the two sets are equal!). Thus the claim follows from Lemma 1.0 ,
Remark. In Cx , every divisor D with Hind u = 0 has the "satellite sequence- c,, := O. For such divisors on Cx , the product theorem holds with (d11 /2). If we set w := z 1 , this is the Weierstrass product E,, i (w (1,7 1 ) for the divisor D' on C with the sequence (d: I )' t. The product theorem 3.1.4 is thus contained in the product theorem above.
ri
"Satellite sequences - (c„),,>1 with c„ E T' or just c„ C C\D do not exist in general: for example, they do not exist for divisors on D := H with However, the following does hold. support T := fi, 2z, 3i, E T : d(T' , > > D, (2) If T' is nonempty and every set T(e) := is finite, then there exists a sequence (c„),,>1 ni T' with 1 1111 il,, — c,d = O.
Proof. Since T' is closed in C, for every d„ there exists c,, E T' such that 1d,, — = d(T' , d,j. If d,, — c„ did not converge to zero. there would exist e0 > 0 such that Id,, — e,, j > eo for infinitely many v. But then the set T( ) ) would be infinite.
If T is bounded and infinite, then T' is nonempty and every set T(E), > O. is finite. (Otherwise some set T(E0), eo > 0, would have an accumulation point (1 4 E T'. which cannot occur since 1d* — wl > d(T' , w) > 6 0 for all w E T(6.0).) Hence: 'Following G. Cantor. we call T' the
derived set of
T in C.
92
4* , Holomorphic Functions with Prescribed Zeros
(3) For every positive divisor D on D with bounded infinite support, there exists a sequence (c„),,>1 in T' with lim Id1, - cu t = O. In particular. it is thus clear that on bounded regions every divisor is a principal divisor (special case of the existence theorem 5). 3. The general product theorem. Let D be an arbitrary region in C. Then, for every positive divisor D on D with support T, there exist Weierstrass products in C\T'. The idea of the proof is to write the divisor D as a sum of two divisors for which there exist Weierstrass products in C\T'. To do this, we need a lemma from set-theoretic topology, which will also be used in 6.2.2 in solving the analogous problem for principal part distributions. Lemma. Let A be a discrete set in C such that A' =
A 1 :=
E A : IzId(A', z)
II,
0. Let
A2 := 1.2" E A : 1.zid(A' z) < 11.
Then A1 is closed in C. Every set A2(E) := 1.z E A2 : d(A' , z) > El, E > O, is finite. Proof. 1) If A 1 had an accumulation point a E C, it would follow that a E A' and there would exist a sequence a„ E A1 with lim a = a. Since d(A', a„) < la - a 11 1, the sequence lanld(A', an) would converge to zero, contradicting the definition of A l . Thus A i = A i . every z E A2 (e) . If there were an eo with A2 (E 0 ) infinite, 2) lz1 < E then A2 (e0 ) would have an accumulation point a E A': but this is impossiEl ble since la - zl > d(A' , z) > fo for all z G A2(60)•
Proof of the general product theorem. We take D to be a positive divisor on 0. Let the sets 211 , T2 be defined as in C\T'. We may assume that T' the lemma (with A := T). Then T; = 0 and 71 = T'. Since T1 and T2 are locally finite in C and C\T', respectively, setting (Z) := D(z) for z E T.
03 (z) := 0 otherwise,
3 = 1,
gives positive divisors D I on C with support T1 and D2 on C\T' with support 2'2. Moreover, D = DI ± 02 in C\T' since T1 n T2 = 0. By the product theorem 3.1.4 there exists a Weierstrass product for D I in C. Since all the sets T2 (E. are finite, 2(2) and the product theorem 2 imply that there exists a Weierstrass product for 02 in C\T'. Hence, by 3.1.2(1), there also exists a Weierstrass product for D = D1 ± 02 in C\T'. )
4. Second proof of the general product theorem. Using a biholomorphic map y, we will first transport the divisor D to a divisor D o v -1 on another region in such a way that a Weierstrass product f exists there for
§1. The Product Theorem for Arbitrary Regions
93
o y -1 ; we will then transport this product back to a Weierstrass prod-
uct f o V for D. We assume that T is infinite, take D to be a divisor on C\T' , fix a E C\-1-1 , and map C\ {a} biholomorphically onto Cx by means of v(z) := (z - a) -1 . Then 0 (1 v(T) and y(T) 1 = v(r). A positive divisor O in C\v(T)/ with support v(T) is defined by i(w) := D(y -1 (:to)),
w E C\v(V),
-6(0) := O.
If (d) >1 is a sequence for D, then (d,),> 1 , with d, := v (d ,,) , is a sequence for d (transporting the divisor by y). Since y(T) is infinite and bounded (because a T), 2(3) and the product theorem 2 imply that
j„, where
L(w) :=
- c,)/(w - c,)] and c„ E v(r),
is a Weierstrass product for D in C\y(V). We now set f,,(z) := f,(v(z)) for z E CVT' U {a}) and set f(a) := 1. Then L is holomorphic in C\T' since limza f,(z) = f(w) = (0) = 1. The normal convergence of fl f„ in C\y(T') implies the normal convergence of 11 f, in CVT' U {a}). Since a is isolated in CV', the product converges normally throughout C\V (inward extension of convergence; cf. 1.8.5.4). Since f,, vanishes only at = v-1 (dv ), and vanishes there to first order, Fl fi, is a Weierstrass product for D in C\T'
5.
Consequences. The product theorem 3 has important consequences for arbitrary regions — as we saw in 3.1.5 for C; the proofs are similar to those of 3.1.5.
Existence theorem. On every region D c C, every divisor is a principal divisor. Factorization theorem. Every function f
0 that is holomorphic in an
arbitrary domain G can be written in the form f=u
fv,
v>i
where u is a unit in the ring 0(G) and FL > , f, is a (possibly empty) Weierstrass product for the divisor (f) in G. In general, the unit u is no longer an exponential function (although it is for (homologically) simply connected domains; cf. 1.9.3.2).
Proposition. (Quotient representation of meromorphic functions). For every function h meromorphic in G there exist two functions f and g, holomorphic in G and without common zeros there, such that h = f g. In particular, the field M(G) is the quotient field of the integral domain 0(G).
4*. Holomorphic Functions with Prescribed Zeros
94
Root criterion. The following statements about a function f E 0(G)\101 and a natural number n > 1 are equivalent: i) There exist a unit u E 0(G) and a function g E 0(G) such that f = ugn ii) Every number o(f), z E G, is divisible by n. In general, the unit u is no longer an nth power. For (homologically) simply connected domains, one can always choose u = 1; cf. 1.9.3.3. In the older literature, the existence theorem was often expressed as follows: Theorem. Let T be an arbitrary discrete set in C and let an integer nd 0 be assigned to every point d E T. Then in the region C\T', where r := T\T, there exists a meromorphic function h that is holornorphic and nonvanishing in (C\T')\T and for which
od(h) = nd
for all d E T.
C\T' is the largest subregion of C in which there exists such a function. Proof. By 2(1), C\T' is the largest region in C in which T is closed. There exists a divisor D on C\T' with support T such that 0(d) = rid, d E T. The existence 0 theorem yields an h E ./14(C\T') with (h) = D.
§2.
Applications and Examples
We first use the product theorem 1.3 to prove that in every integral do-
main 0(G) there exists a greatest common divisor for every nonempty set. We then deal explicitly with a few Weierstrass products in E and C\OE, including a product due to E. Picard, which is constructed with the aid of the group SL(2,Z). 1. Divisibility in the ring 0(G). Greatest common divisors. The basic arithmetic concepts are defined in the usual way: f E 0(G) is called a divisor of g G 0(G) if g = f • h with h E 0(G). Divisors of the identity are called units. A nonunit y 0 is called a prime of 0(G) if y divides a (finite) product only when it divides one of the factors. The functions z c, c E G, are — up to unit factors = precisely the primes of 0(G). Functions 0 in 0(G) with infinitely many zeros in G cannot be written as the products of finitely many primes. Since such functions exist in every domain G by Theorem 1.3, we see: No ring 0(G) is factorial.
Despite this, all rings 0(G) have a straightforward divisibility theory. The reason is that assertions about divisibility for elements g 0 axe
§2. Applications and Examples
95
equivalent to assertions about order for their divisors (f), (g). Writing D < -6 if -6 — D is positive, we have the simple Divisibility criterion. Let f, g E 0(G)\{0} . Then
f divides g (f) _< (g). Proof. f divides g if and only if h := glf E 0(G). But this occurs if and only if oz (h) = oz (g)—o z (f) .? 0 for all z E G, i.e. if and only if (f) < (g). ID If S is a nonempty set in 0(G), then f E 0(G) is called a common divisor of S if f divides every element g of S; a common divisor f of S is called a greatest common divisor of S if every common divisor of S is a divisor of f. Greatest common divisors — when they exist — are uniquely determined only up to unit factors; despite this, one speaks simply of the greatest common divisor f of S and writes f = gcd(S). A set S 0 0 is called relatively prime if 1 = gcd(S).
S 0 0 is relatively prime if and only if the functions in S have no common zeros in G, i.e. if ngEs z(g) = 0 . The proof of the next result is a simple verification. If f = ged(S) and g = gcd(T), then gcd(S UT) = gcci{ f,g}.
0 is a set of positive divisors
D on G, then the map G —. Z, z ,--+ min{0(z) : 0 E D} is a divisor min{D : D E Z} > O. The divisibility If 0 0
criterion implies the following: Every function f E 0(G) with (f) = minf(g) : g E S, g 0 01 is a gcd of S 0 {0}. The next statement is now an immediate consequence of Theorem 1.3. Existence of the gcd. In the ring 0(G), every set S 0 0 has a gcd. Proof. Given S 0 {0}, choose f E 0(G) such that (f) = minf(y) : g E 0 S, OE
It may seem surprising that the product theorem 1.3
is needed to prove the
existence of the gcd (even if S has only two elements!). But it should not be forgotten that there are integral domains with identity in which a gcd does not always exist: in the ring Z[V-L-g], for example, the two elements 6 and 2(1+ V7-75.) have no gcd. In principal ideal rings such as Z, Z[i], and C[z], every set S has a gcd; in fact, it is always a finite linear combination of elements of S. This assertion also holds for the ring 0(G) if S is finite, as we will see in 6.3.3 by means of Mittag-Leffler's theorem. Exercises. Define the concept of the least common multiple as in number theory and, if it exists, denote it by lcm. Prove:
4* • Holomorphic Functions with Prescribed Zeros
96
0 has a least common multiple. If (f) = lcrn(S) 0, then I) Every set S (f) = max{ (g) : g G S } . 2) If f and g are respectively a gcd and lcm of two functions u, V E 0(G)\ {0}. then the products f g and u • y differ only by a unit factor.
2. Examples of Weierstrass products. 1) Let D > 0 be a divisor on E that satisfies D(0) 0 and has sequence (d„),>1, and suppose that E v>1 (1 - Idy l) < oo. Then
(*) H E0 v>i
(d,
-1
z -
)
= H d,_
- d,
v>1 d„z - 1
- -1 -
E 0(C\{di
, d2 • • • •})
is a Weierstrass product for D. Proof. Let c, := 1/4; then
E icy
- d. < oc since
= 141' (1 — 141 2 ) 5_
(+) where in k, := O.
2m-i (1 - Idy l),
minfld„I : v > 11. The assertion follows from Lemma 1.1 with D
The products (*) are bounded in E; up to a normalization, they are Blaschke products (cf. 3.3). Because of their importance, we also give a direct proof of convergence. It follows from (+) that
a
-
dz
-
d 1
1- 1d1 2
1
1 (1 1 I z
t for all z E K and all u > 1. Thus
t > 0 such
-
z - d,,
2t -1
1 K
Ii>,
— v>1
- -t - - 1 which implies the normal convergence of (*) in CVd 1 ,d2 • •}.
2) Let r„ > 0, r,, 1, be a sequence of pairwise distinct real numbers with limn, = O. The set
T := {d„p := (1 - r,)c,p , 0 < p < u , ii = 1,2,• • •}, where
c„p
exp(2/nr/v)
E
8E, is locally finite in c\alE. Since dvp
tends to 0, the product theorem 1.2 implies that oc
ti
-i
H Ev_,
,= 1 p=0
ev„ _ z
-
c„p =
§2. Applications and Examples
97
is a Weierstrass product in C\OE, which vanishes to first order precisely at the points of S. = vr 1 converges in the cases Since dLip r, = 1/v, k = 2 and r, = 1/v3 , k = 0, the corollary to Lemma 1.1 gives the Weierstrass products
E=E
H (1 + v,p
fi V,p
1
(
Cup
VZ
(i + 1
)
.2
exp [
Cup cup
,3 z c vp
)
,
Cl/i)
v(c „p
-
z)
+
(
vp
.
21)2 (c,p - z) 2
and
respectively.
3. On the history of the general product theorem. 'Weierstrass left it to others to extend his product theorem to regions in C. As early as 1881 ([Pi], pp. 69-71), E. Picard considers the region C\OIE; lie discusses, among other things, the product
B)
HEi
z where A :=
z-A A-B =ll z_B exp z_B ,
+ - (a -
B := + 62:
1. and a, 0, -y, and 6 run through all numbers in Z satisfying al) Picard's product is probably the first example of a Weierstrass product in a region C where convergence-producing factors of the Weierstrass type were consciously used. Picard said nothing about the convergence of his product, but in 1893 made the following comment (cf. Traite d'analyse, vol. 1, P. 149): "... , c'est ce que l'on reconnaît en considérant à la place de la série une intégrale triple convenable dont la valeur reste finie quand les limites deviennent infinies." (... , it is what one recognizes if, instead of the series, one considers an appropriate triple integral whose value remains finite when the limits become infinite.) A year later . Picard studied slit, regions ([Pi], pp. 91-93). He introduced the products - c,,)/(z c,)) in his notes. They were also used in 1884 by Mittag-Leffler to prove the existence theorem for general regions ([ML], especially pp. 32 38). Picard's notes are not mentioned by Mittag-Leffler; in 1918 Landau ([4 P. 157) speaks of the "well-known Picard-Mittag-Leffler product construction." In their work [BStd. carried out in 1948 but not published until 1950, H. Behnke and K. Stein extended the existence theorem 1.5 to arbitrary noncompact Riemann surfaces (loc. cit.. Satz 2, p. 158).
11
4. Glimpses of several variables. With his product theorem, Weierstrass opened the door to a development that led to new insights in higher-dimensional function theory as well. The product theorem was generalized to the case of several complex variables as early as 1895 by P. Cousin, a student of Poincaré. in
98
4*. Holomorphic Functions with Prescribed Zeros
[Co]. The formulation of the concept of divisors already presented difficulties at this point, since the zeros of holomorphic functions in C'n , n. > 2, are no longer isolated but form real (2n - 2)-dimensional surfaces. Cousin and his successors could derive the analogous theorem only for Cn itself and polydomains in Cn (product domains G I x G2 X • • • X an , where each G, is a domain in C). Cousin thought he had proved his theorem for all polydomains. But the American mathematician T. H. Gronwall discovered in 1917 that Cousin's conclusions hold only for special polydomains; at least (n - 1) of the n domains G1, , Gn must be simply connected (cf. [Gro ], p. 53). Thus there exist -- and this was a sensation - topological obstructions! It was soon conjectured that Cousin's theorem was valid for many topologically nice domains of holornorphy; 2 for example, H. Behnke and K. Stein proved in 1937 that the theorem holds for all star-shaped domains of holomorphy UBSt 2 1, p. 188). The Japanese mathematician K. Oka achieved a breakthrough in 1939; he was able to show that, in arbitrary domains of holotnorphy G c C" , a positive divisor is the divisor of a function holomorp hic in 61 if and only if it is the divisor of a function continuous in G ([0], pp. 33-34). This statement is the famous Oka principle, which K. Stein generalized in 1951 to his manifolds, interpreted homologically, and made precise [St]. It was .I-P. Serre who. in 1953, gave the final solution of the Cousin problem ([S], pp. 263-264): In a Stein manifold X, a divisor D is the divisor of a function meromorphic ni X if and only if its Chem cohomology class c(0) E H 2 (X,Z) vanishes. In particular, on a Stein manifold X with H 2 (X, Z) 0, every divisor is a principal di vis OT . The significance of the second cohomology group with integer coefficients for the solvability of the Weierstrass-Cousin problem becomes evident here. In the thirties, the fundamental group 7r (X) was still thought to have great importance in this context: but in [S] (p. 265), Serre exhibited a simply connected domain of holomorphy in C3 where not all divisors are principal divisors. The methods that Serre and Cartan developed jointly to prove Serre's theorem revolutionized mathematics: the theory of coherent analytic sheaves and their cohomology theory began their triumphant advance. Serre ([S], p. 265) also put the finishing touches on Poincans's old theorem (cf. 3.1.6): In a Stein manifold, every m.eromorphic function is the quotient of two halomorphic functions (which need not be locally relatively prime). We must be satisfied with this sketch; a comprehensive presentation can be found in [CRI. The Oka principle was substantially extended by H. Grauert in 1957; he showed, among other things, that holomorphic fiber bundles over Stein manifolds are holomorphically trivial if and only if' they are topologically trivial ([Gra], p. 268 in particular). Weierstrass, Poincaré, and Cousin would certainly have been impressed to see how their theories culminat4d in the twentieth century in the Oka-Grauert principle: Locally prescribed analytic data with globally continuous solutions always have globally holomorphic solutions as well.
2
For the concept of domains of
holomorphy and Stein manifolds, see also 5.2.6.
§3. Bounded Functions on E and Their Divisors
§3.
99
Bounded Functions on E and Their Divisors
By the product theorem 1.3, the zeros of functions holomorphic in D call be arbitrarily assigned in D as long as they have no accumulation point there. The situation changes when growth conditions are imposed on the functions. Thus, for divisors 0 on C. there never exist bounded functions f with (f) = D. < 1 for A Weierstrass product 11 f, is certainly bounded in D if all v. Products with such nice factors are rare. In what follows, we study the case D = E. For the functions z— d gd(z) =
1
, d E E,
which we recognize as automorphisms of E, Igd I E = 1. We will see that a divisor D > 0 on E with 0(0) = 0 is a divisor of a bounded function on E if and only if D has a Weierstrass product of the form H ( d,/ dv )gd,,(z), and that such products exist if and only if
_ Idyl) < cx (Blaschke condition). The necessity of this condition, which implies an identity theorem, follows quickly (in Subsection 2) from Jensen's inequality (Subsection 1). Its sufficiency is proved in Subsection 3. 1. Generalization of Schwarz's lemma. Let f G 0(E), and let E E be porwise distinct zeros of f. Then (1)
If(z)1
z — d1
z — d,
a„z — 1
z —1
I fI
,d,,
for all z E E.
I, kin 1}. Set h := Proof. Let z E E be fixed, and let ni := max{ principle implies that the maximum 11794; then g := f lh E 0(E), arid Ig(z)I If/ min 1 „, 1=7.11h(w)11 for all r E (m, 1). Now I h(w)1 = 1 for all ,
E OE (since gd(w) = (1 — (170)1(71w — 1) for w E 0E). Hence
lim min flh(w) 11 = 1, I. kv i,
and thus
Ig(z)1
1 If.
Remark. If IfiE < 1 and n = 1, d i = 0. (1) is Schwarz's lemma; cf. 1.9.2.1. As in that case, we now have a sharpened version of the result:
If equality holds in (1) for a point d E
f (z) =
H
gdv(Z) ,
dnI,
with ri E
then
100
4*. Holoinorphic Functions with Prescribed Zeros
For z = 0, (1) becomes "Jensen's inequality": (2)
I f(0)1 < Id1d2
• dni
flE•
Inequality (2) is a special case of Jensen's formula, which we derive in the appendix to this section. Historical note. The proof above dates back to C. Carathéodory and L. Fejér [CFI. Inequality (2) appears in the work of J. L. W. V. Jensen ([J], 1898 -99) and of J. Petersen ([13], 1899). 2. Necessity of the Blaschke condition. Let f 0 be holomorphzc and bounded in E, and let d1, (12 ,... be a sequence for the divisor (f). Then
E(1 -141) < oc. Proof. We may assume that f(0) 0. Then E(1 - Id y l) = oc would imply that lim Id 1 d2 d„I = 0 by 1.1.1(d), and hence f(0) = 0 by 1(2).
As a corollary, we point out the surprising Identity theorem for bounded functions on E. Let A = be a countable set in E such that E(1— ja„1) = oc. Suppose that f, g are bounded in E and flit = glA. Then f = g.
e 0(E)
Proof The function h := f - g E 0(E) is bounded in E. If h were not the zero function, then E(1 —1aI) would be a subseries of the series E(1 — Idy l) determined by the sequence (d11 ) for the divisor of h. Hence it would be D convergent by the statement above.
Bounded holomorphic functions in E therefore vanish identically as soon as their zeros move too slowly toward the boundary of E (this is made precise by E(1 - ja,1) = Do). Thus f E 0(E) must be the zero function if it is bounded and vanishes at all points 1 - 1/n, n > 1. 3. Blaschke products. For each point d E E. we set
(1)
b(z, d) := b(z , 0)
z-d _ d dz - 1 =1d1-1 E0
d z-d1
if d 0,
:= z.
Then b(z, d) is holomorphic on E. and nonvanishing in E\ {d}, the point d is a first-order zero, and b(z , = 1. Now let D > 0 be a divisor on E and let (d11)v>1 he a corresponding sequence. The product
b(z) := H b(z , d„) y>1
§3. Bounded Functions on E and Their Divisors
101
is called the Blaschke product for D if it converges normally in E (and hence even in c\aE). Blaschke products are thus a special class of Weierstrass products. (2) If b is a Blaschke product, then b E 0(E), (b) = D. and IblE < 1. Moreover, if b(0) 0, then 00
b(z) = b(0) -1 H EoRd, — v.t
- d„ 1 )b
with
b(0) :=
Example 1) in Subsection 2.2 contains the existence theorem for Blaschke products: 00 Idui) O on E are
equivalent: i) D is the divisor of a function in 0(E) that is bounded in E. ii) ED(z)(1 — 121) < oc (Blaschke condition). zE E 00
iii) The Blaschke product H b(z,cl i,) exists for D. 1.,=1 Because of ii) there does not exist, for example, any bounded function f E 0(E) that vanishes to nth order at each point 1 - 1/n 2 , n E NVOI. The following is immediate. For every bounded function f E 0(E), there exist a Blaschke product b and a. function g E 0(E) such that f = eg h.
Historical note. W. Blaschke introduced his products and proved the existence theorem in 1915 ([B11, p. 199). Of course — as the title of his paper indicates — Blaschke was then mainly interested in Vitali's convergence theorem; we go into this in 7.1.4. Edmund Landau reviewed Blaschke's work in 1918 and simplified the proof by using Jensen's inequality; cf. [LI.
4* • Holomorphic Functions with Prescribed Zeros
102
4. Bounded functions on the right half-plane. Setting t(z) := (z-1)/(z+1) gives a biholomorphic map of T := {z E C : Re z > 0} onto E; we have
(1)
1 - It(z)1 -2
= 4 Re z 4 Re(1/z) lz + 11 2 11 + z-112
for all z E CVO, -1}.
The results obtained for E carry over easily to T.
a) A positive divisor D on T with corresponding sequence d1, d2, . .. is the divisor of a bounded holomorphic function on 7 if and only if
x--, Re ds, L. i 11 + d1 e, 2 < 1
(Blaschke condition for T).
Id=
b) Let the function f E 0(T) be bounded in T and vanish at the pairurise distinct points d,, d2, . . ., where 6 := inf{idn i} > 0 and Re (1/d,,) = Do. Then f vanishes identically on T. Proof. a) The map D o F' : E --0 ni is a positive divisor on E with corresponding sequence C-4, := t(d„). If f E 0(T) is bounded in 7, then (f) = D if and only if (f o t - ') = 0 o t', where fo t -1 E 0(E) is bounded in E. By Proposition 3, this occurs if and only if E(1 - IiiI) < cc. The assertion now follows from (1) because
1 - (1 - lw1 2 ) 5_ 1 -1w1 < 1 -1w1 2 for all w E E. 2
b) Since 11 + w - '1 -2 > (1 + 6 -1 ) -2 for all w with 1w1 > 6, it follows from (1) that
E
x-, Re dy 1 (1 Re (1/4) = co. +61)2 L-s 11 + d12 i, _>. v>i p>1 0
By a), f must then vanish identically.
Statement b) will be used in 7.4.3, in the proof of Miintz's theorem. Analogues of a) and b) are valid for the upper half-plane H. The map H Z 7, z 1 + iz, is biholomorphic and Re (-iz) = Im z. In this situation, we thus obtain the convergence condition E (1 mdidii + 412) < oc in a) and the divergence condition E Im(1/4) = -oc in b). -
-
Exercise. Define "Blaschke products" for T and H and prove the analogue of 3(3) for these half-planes.
Appendix to Section 3: Jensen's Formula
Jensen's inequality
3.1(2)
can be improved to an equality:
Jensen's formula. Let f E 0(E), f(0) p. Let 0 < r < 1 and let d1, d2 ,... ,d,„ be all the zeros of f in Br (0), where each zero appears according to its order. Then // r
(J)
n
log I f (0)1 + log 164(12 . . . d n 1 -=
:7r rir log If (re29)40.
§3. Bounded Functions on E and Their Divisors
103
The integral on the right-hand side is improper if f has zeros on the boundary of Br (0). The second summand on the left-hand side is zero if f has no zeros in Br (0). Since logx is monotone for s > 0, (J) immediately leads to the inequality rn If (0)1 5_ Iclid2
IfloBr (o)-
Jensen's inequality 3.1(2) follows by passing to the limit as r
1.
We reproduce the proof given by J. L. W. V. Jensen in 1898-99 ([ 4 p. 362 ff.); he also admitted poles of the function f. The formula can also be found in an 1899 paper of J. Petersen ([Pe], p. 87). We write B for B(0). Our starting point is the following special case of
(J) (1) If g E 0(E) has no zeros in B, then loglg(0)1 =
1
27r
fo log ig(re )09.
Proof. There exist a disc U with BCUCE and a function h E 0(U) such that gIU = g(0) exp h with h(0) = 0 • 3 Since h(z)/z E 0(U), h(()
0= f
2r
= z f h(re )cla.
&B
0
Since Re h(z) = loglg(z)Ig(0)1 for z E U, it follows that 277 0 =
27r
Re f h(rencle = I loglg(rez ° )40 - 27r loglg( 0)1. Jo
Statement (1) is Poisson's mean-value equation for the function log Ig(z)l, which is harmonic in a neighborhood of B; cf. also 1.7.2.5.
In order to reduce (J) to (1), we need the following result.
10
(2)
27r
log 1 - e 6 1c10 = O.
Proof. Since 11 - e2i(PI = 2 sin ço for cp E [0,7], we have (with 9 = 2)
1/27r log II - e''
20
= f log(2 sin (p)dp = r log 2 + f log sin (p&p.
U such that gl LI has no zeros. Since U is star-shaped, g E 0(U). Then set h:=Tt Ît(0). 3 Choose
—
exp it with
104
4* • Holomorphic Functions with Prescribed Zeros
The integral on the right - hand side exists and, by 1.3.2(1), equals - 7r log 2.
Formula (2) is usually derived by function-theoretic methods. The direct calculation above supports Kronecker's sardonic maxim on the occasionally "gute Friichte bringenden Glauben an die Unwirksamkeit des Imaginaren" (fruitful belief in the inefficacy of the imaginary); cf. also 1.14.2.3. It is now easy to prove (J). If c l , , cm are all the zeros of f on function n — g(z) du z - r 2 in cA f(z) E 0(E) 11 r(z - du ) 11 c u -z v=i
has no zeros in B. Since g(0) = f(0)rn/d 1 d2 ...4 and zE
aB,
aB, the
dv z - r2 = 1 for r(z - dv )
it follows from (1), if we set cp = re i°,4, that
log If (0)1 + log (*)
27r
Idid2 • .
2ir
1
=
Tn
0
In
log f(re i9 )
(1 _
dO.
ti •=.1
Since the integrand on the right-hand side is the difference log If (rei° )I E p1 (J) follows from (*) because of (2). =m log 11 -
-
In applications, r can often be chosen so that f has no zeros on OB,(0) (as in the derivation of 3.1(2), for example). Then the factors cp /(ci, - z) drop out, and (J) follows directly from (1) — without using (2).
Jensen's formula has important applications in the theory of entire functions and the theory of Hardy spaces; for lack of space we cannot investigate these further.
Bibliography [BSti]
BEHNKE, H. and K. STEIN: Elementarfunktionen auf Riemannschen Flachen , Can. iciurn. Math. 2, 152-165 (1950).
IBSti]
BEHNKE, H. and K. STEIN: Analytische Funktionen mehrerer Veranderlichen zu vorgegebenen Null- und Polstellenflachen, Jber. DMV 47, 177-192 (1937).
[Bil
BLASCHKE, W.: Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, Ber. Verh. Kiinigl. $àchs. Ces. Wzss. Leipzig 67, 194-200 (1915); Ces. Werke 6, 187-193.
[CF]
CARATHÉODORY, C. and L. FEJÉR: Remarques sur le théorème de M. Jensen, C.R. Acad. Sci. Paris 145, 163-165 (1907); CARATHgODORY'S Ges. Math. Schrift. 3, 179-181; FE.IgR'S Ces. Arb. 1, 300-302.
§3. Bounded Functions on E and Their Divisors
105
P.: Sur les fonctions de n variables complexes, Acta Math. 19, 1-62 (1895).
[Co]
COUSIN,
PR]
GRAUERT, H. and R. REMMERT: Theory of Stein Spaces, trans. A. HUCKLEBERRY, Springer, New York 1979.
[Gra]
GRAUERT, H.: Analytische Faserungen über holomorph-vollstdndigen Rüumen, Math. Ann. 185, 263-273 (1958).
[Gm]
GRONWALL, T. H.: On the expressibility of a uniform function of several variables as the quotient of two functions of entire character, Trans. Amer. Math. Soc. 18, 50-64 (1917).
[J]
JENSEN, J. L. W. V.: Sur un nouvel et important théorème de la théorie des fonctions, Acta Math. 22, 359-364 (1898-99).
[L]
LANDAU. E.: Über die Blaschkesche Verallgemeinerung des Vitalischen Satzes, Ber. Verh. Kiinigl. Siichs. Ges. Wiss. Leipzig 70, 156-159 (1918); Coll. Works 7, 138 141.
[ML]
MITTAG-LEFFLER, G.: Sur la représentation analytique des fonctions monogenes uniformes d'une variable indépendante, Acta Math. 4, 1-79 (1884). OKA, K.: III. Deuxième problème de Cousin, Journ. Set. Hiroshima Univ., Ser. A, 9, 7-19 (1939); Coll. Papers, trans. R. NARASIMHAN, 24-35, Springer, 1984.
[Pe]
PETERSEN. J.: Quelques remarques sur les fonctions entières, Acta Math. 23, 85-90 (1899).
(Pi]
PICARD, E.:
[S]
SERRE, J-P.: Quelques problèmes globaux relatifs aux variétés de Stein, Coll. Fonct. Plus. Var. Bruxelles 57-68 (1953); Œuvres 1, 259-270.
1St]
STEIN. K.: Analytische Funktionen mehrerer komplexer VerAnderlichen zu vorgegebenen Periodizitasmoduln und das zweite Cousinsche Problem, Math. Ann. 123, 201-222 (1951).
Œuvres 1.
5 Iss'sa's Theorem. Domains of Holomorphy
We begin by giving two interesting applications of the Weierstrass product theorem that have not yet made their way into the German textbook literature. In Section 1 we discuss Iss'sa's theorem, discovered only in 1965; in Section 2 we show — once directly and once with the aid of the product theorem — that every domain in C is a domain of holomorphy. In Section 3 we conclude by discussing simple examples of functions whose domains of holomorphy have the form tz E C: lq(z) 1 < RI, q E C[z]; Cassini domains, in particular, are of this form.
§1. Iss'sa's Theorem --+ G between domains in C Every nonconstant holomorphic map h : lifts every function f meromorphic in G to a function f o h meromorphic in G. Thus h induces the C-algebra homomorphism : lvf(G) —) M(6), f
f 0 h,
which maps 0(G) into 0(6) (cf. also 1.10.3.3). Iss'sa's theorem says that every C-algebra homomorphism .A4 (G) —) M(6) is induced by a holomorphic map d -+ G. In preparation, we prove that all C-algebra homomorphisms 0(G) —) 0(0) are induced by holomorphic maps a, G. The proof of this theorem of Bers is elementary; it is based on the fact that every character X: 0(G) —) C is an evaluation. The proof of Iss'sa's general theorem, however, requires not only the Weierstrass product theorem but also tools from
108
5. Iss'sa's Theorem. Domains of Holomorphy
valuation theory; in the background is the theorem that every valuation on .A4(G) is equivalent to the order function oc of a point c E G (Theorem 5). -- G and C always denote domains in C.
1. Bers's theorem. Every C-algebra homomorphism 0(G)
C is called C, f a character of 0(G). Every evaluation x, : 0(G) 1(c), c E G, is a character. We prove that these are all the characters of O(G). (1) For every character x of 0(G), x = x c with c := x(idG) E C . Proof. Set, e(z) := z — c; then x(e) = x(idG) — c = 0. It follows that c E G, since otherwise e would be a unit in 0(G) and we would have 1 = x(c • c - I) = x(e)x(e -1 ) = 0. Now let f E O(G) be arbitrary. Then f (z) = f (c) + e(z) f (z), with fi E 0(G). It follows that
X(f = )
X f(c) (
)
+ X(e)X(ft) = f (c) =
Xe(f ):
hence
x = Xr•
The theorem now follows quickly from (1).
Bers's theorem. For every C-algebra homomorphism : O(G) 0(d), there exists exactly one map h G such that (p(f) = f o h for all f E 0(G). In fact, h = cp(idG) E 0(d). — (t9 is bijective if and only if h is Inholornorphic. Proof. Since h is to satisfy ( f) = f o h for all f, it must satisfy (p(idG) = idG 0 h = h. We show that the theorem does in fact hold for h := ci)(idG). Since x„ o p, a G 6, is always a character of 0(G), it follows from (1) that Xa ° = Xe,
with
c = (xa 0 c,o)(idG) = x a (h) = h(a),
a E C.
Hence (p(f) = f o h for all f E 0(G), since we now have
4(f)(a) = xa(So(f)) = (Xa o p)(f) = xh(„)(f) = f(h(a)) = (f o h)(a) for all a E Ô. The last statement of the theorem is immediate.
Bers's theorem contains some real surprises: — if the function algebras 0(G) and 0(d) are algebraically isomorphic, then the domains G and 6' are bihdlarnorphically isomorphic; 0(C;) is automatically — every C-algebra homomorphism p : 0(G) continuous (if a sequence in 0(G) converges compactly in G to f, then the image sequence converges compactly in G to (p(f)).
§1. Iss'sa's
Theorem
109
2. Iss'sa's theorem. Let yo : M(G) .M(d) be any C-algebra homomorphism. Then there exists exactly one holomorphic map h :G -+ G such that p(f) = f o h for all f E M(G).
Because of Bers's theorem and because M(G) is the quotient field of 0(G) (cf. 4.1.5), it suffices to prove the following lemma. Lemma. For every field homomorphism ( to : M(G) M(d),
c,o(0(G)) C 0(6).
The proof is carried out in the next subsection. We use methods from valuation theory (well known to algebraists, but less familiar to classical function theorists). We write M(G)X for the multiplicative group M(G)\{O}. Z is called a valuation on. .A.4(G) if for every f, A map y : M(G)x B1) v( f g) = v(f) + v(g) B2) v(f + g)
(product rule),
min{v(f),v(g)}
if f 0 —g.
Our next result is immediate. If v is a valuation on M(G), then v(c) = 0 for all c E C x . Proof. For every n > 1 there exists cn E C x such that (c„)n = c. It follows from B1) that v(c) = nv(c„) E nZ for all n > 1; but this is possible only if
v(c) = 0. Condition B2) can be sharpened:
B2') v(f + g) = min{v(f), v(g)}
if f 0 —g and v(f) v(g).
Proof. Let v(f) v(g). Since v(—g) = v(g), it follows from B2) that v( f) > min{v(f + g) v(g)).
min{v(f),v(g)} =
thus min{v(f + g), v(g)} = v(f) . Hence v(f + g) = v( f) if v(f) < v (g) The valuations on M(G) that are important in function theory are the order functions oc , c E G, which assign to each function f E M(G)x its order at the point c; see, for example, 1.10.3.4. The following is immediate: Holomorphy criterion. A meromorphic function f E morphic in G if and only if oc (f) > 0 for all c E G.
M WY' is holo-
3. Proof of the lemma. The core of the proof is contained in the next
lemma.
110
5. Iss'sa's Theorem. Domains of Holomorphy
Auxiliary lemma. If v is a valuation on M(C), then v(z) > 0. Proof (cf. [Is), pp. 39-40). Suppose that v(z) = -rn with m > 1. Since v(c) = 0 for all c e Cx , it follows from B2') that
(1)
v(z -
=
for all c
Cx .
Now let d E N, d > 2. By the existence theorem 3.1.5, there exists a function E 0(C) which has no zeros in C\N and vanishes to order dk at k E N. Set qn (z) := q(z)
ITE -1 (z - v) d'' , n > 1.
Then qn E 0(C). and B1) and
1) imply that n-1
(2)
v(g„) = v(q) +
4-" cT
d" = v(g) +
m
cl - 1
(d" - 1).
By the construction of q„, rin divides every number o.,„(q„), z E C; hence, by the root criterion 3.1.5, there exists g, E 0(C) with gncin = q„. Thus d" v(g„) = v(q) and. by (2), (3)
v(q) +
rn, (dn - 1) E d"Z d-1
for all
71, >
1.
Therefore (d - 1)v(g) - in E criZ for all n > 1, which is possible only for v(q) = rni(d - 1). Since d > 2 was arbitrary, this gives the contradiction 7"71 -= O. The next statement is immediate from the auxiliary lemma: (*) If v is any valuation on M(G), then v(f) > 0 for all f G 0(G)\{0}. 0 in 0(G), the map Proof of (*). A verification shows that for every f X Z, g 1-• v(go f). is a valuation on M(C). Since v1 (z) = v(f), 7? f : .A4(C) (*) follows from the auxiliary lemma. After these preliminaries, the proof of the lemma is easy: Since w is 0 for all f E ./t4 (C ) ' . Hence. injective as a homomorphism of fields, w( f) for every c E G. setting
vc(f) := 0, ( P(f)), f E M(G)x
,
defines a valuation on M(G). By (*), 0(7(f)) > 0 for all c E Ô if f E 0(G) x . The holomorphy criterion 2 then implies that yo(f) E 0(6), and the assertion follows. 4. Historical remarks on the theorems of Bers and Iss'sa. The American mathematician Lipman Bers discovered his theorem in 1946 and
§1. Iss'sa's Theorem
111
published it in 1948 [Ber]. Bers considers only isomorphisms; he works with the maximal principal ideals of the rings 0(G) and 0(6). Incidentally, Bers proves more: he proceeds from ring isomorphisms : 0(G) 0(e';') and shows ingeniously. that ço induces either the identity or conjugation on C; h: G is correspondingly biholomorphic or anti-biholomorphic. Before Bers, C. Chevalley hnd S. Kakutani had already studied the difficult case of the algebra of bounded holomorphic functions (unpublished). A historical survey can be found in [BuSa] (p. 84). Bers's theorem is also valid for holomorphic functions of several variables, if their domains of definition are assumed to be normal Stein spaces. But the proof becomes rather demanding: one must use cohomological methods and resort to the theory of coherent analytic sheaves (cf. [GR], Chapter V, §7).
Hej Iss'sa (the pseudonym of a. well-known Japanese mathematician) extended Bers's theorem to fields of functions in 1965. He immediately handles the case of complex spaces (cf. 2.6); his result is the following ([I], Theorem II, p. 34): Let G be a normal complex space and G a reduced Stein space, and let .A4(6) be any C-algebra homomorphism. Then there exists : M(G) G such that cp(f) = f o hfor all exactly one holomorphic map h : E M(G). Once again, the hard part of the proof is to show that p maps the ring 0(G) into 0(G- ). Iss'sa's theorem should also be compared with the 1968 paper [Ke] of J. J. Kelleher.
M(G). The algebraically inclined reader will ask whether there actually exist valuations on .A4(G) that are not order functions. Certainly the function MOc is a valuation on M(G) for every point c E G and every rn E N. We prove that there are no other valuations. 5*. Determination of all the valuations on
0 on ,A4(G), there exists exactly one pond Proposition. For every valuation v c E G such that v(z — c) > 1. Moreover, v(h) = moe (h) for all h E where m := v(z — e). Proof (cf. [I], pp. 40-41). First, v(e) = 0 for every unit e E 0(G). since 0 = v(1) = v(1/e) and since both v(e) > 0 and 7:(1/e) > 0 hold by 3(*). v(e • 1/e) = v(e) We now set A := {a E G : v(z — a) > 0} and claim that (#) v(f) = 0 for every f E 0(G)\ {0} such that Z(f)
nA
= 0.
If Z ( f) is finite, then f (z) = e(z) nun = , (z - c,,), where e is a unit in 0(G). Since v(z — c„) > 0 by 3(*) and since cv fl A. it follows from B1) that v( f ) = v(e) =- 0. On the other hand, if Z( f) = te i , e2. .1 is infinite, we use the existence theorem to choose h E 0(G) such that 4.15
Z(h) = Z(f)
and
oe ,,(h) = oc ,,(f) • (v! — 1), v = 1, 2, .
112
5. Iss'sa's Theorem. Domains of Holomorphy
For hn := h • f I 1-17: 1 (z -
Z(h n ) =
E 0(G), we then have
v(h n ) = v(h)+v(f), and 0,, (hn) = oc u (f )- v! for v > n.
Therefore n! divides every number o(h), z E G; thus, by the root criterion 4.1.5, there exists gn E 0(G) such that h n /giT is a unit in 0(G). It follows that v(h) = n!v(g n ) and hence that v(h) + v(f) = v(h) E n!Z, n = 1, 2, .... This implies that v(h) + v(f) = 0. Since v(h) > 0 and v(f) > 0 by 3(*), we obtain v(f) = 0. This proves (#). It follows immediately from (#) that A is nonempty, since otherwise v(f) = would hold for all f E 0(G)\{0}, and therefore — since, by 4.1.5, M(G) is the quotient field of 0(G) — y would have to be the zero valuation. Hence there exists some c E A. There exist no other points c' E A, c' c, since the equation r(z - c') - r(z - c) = 1, where r := (c - c') -1 E C x , would lead to the contradiction 0 = v(1) > minlv(z - c'),v(z - c)} >0. Thus A = {c}, proving the first statement of the proposition. Now let m := v(z c). Then, if f 0 is in 0(G) and n := o(f), the function g := f 1(z - c) n E 0(G) has no zero in A. It follows from (#) that v(g) = 0,
i.e. v(f) = v((z
On) = moc (f).
Taking quotients gives v(h) = mo c (h) for all h E .A4(G) >< .
§2.
Domains of Holomorphy Es giebt analytische Functionen, die mir far einen Theil der Ebene existieren und far den abrigen Theil der Ebene gar keine Bedeutung haben. (There are analytic functions which exist only for part of the plane and have no meaning at all for the rest. of the plane.) — Weierstrass, 1884
1. A domain G in C is called the domain of holomorphy of a function f holomorphic in G if, for every point c E G, the disc about c in which the Taylor series of f converges lies in G. The following is immediate: If G is the domain of holomarphy of f , then G is the "maximal domain of existence" of fi in other words, if G D G is a domain in which there exists a function f E 0(6) such that fiG = f, then d coincides with G. If a disc is the maximal domain of existence of f, then it is also the domain of holomorphy of f (prove this); the definition given in 1.5.3.3 for discs is thus consistent with the definition above. In general, however, domain of
§2. Domains of flolomorphy
113
holomorphy means more than maximal domain of existence. For example. the slit plane C - is the maximal domain of existence of the functions \rz and log z E 0(C- ) but not their domain of holomorphy: the Taylor series for V-Tz and log z about c E C- have Bi ci (c) as their disc of convergence, and B1 (e) C - if Re c < 0. (The functions and log z can be continued holomorphically "from above and below" to every point on the negative real axis, but all the boundary points of C- are "singular" for VZ and log z, in the sense that none has a neighborhood U with a function h E 0(U) that coincides in U n C - withV.Z or log z; cf. 1.5.3.3 and Subsection 3 of this section.) 2. The domains C, Cx , and E are the domains of holomorphy of z, z 1 , and z 2v , respectively (for the last example, cf. 1.5.3.3). The focal point of this section is the following general theorem.
E
Existence theorem. For every domain in C, there exists a function f holomorphic in G such that G is the domain of holomorphy of f. There are two ways to prove this. A function f E 0(G) is constructed that either tends to oc as the boundary of G is approached or has zero set Z(f) G accumulating at every boundary point. Difficulties arise if the boundary OG is tricky (if it has accumulations of spikes, for example, as in Figure 8.4, page 176). We must ensure that the boundary is approached "from all directions inside G." To define this kind of approach to the boundary, we introduce the concepts of well-distributed boundary set and peripheral set. The first proof then goes through with "Goursat series"; the second exploits the existence theorem 4.1.5. In what follows, we use concepts and methods of proof from set-theoretic topology. We use the fact that every point of an open set D in C lies in a uniquely determined connected component of D (cf. for example 1.0.6.4), which for brevity we call a component of D; every such component is a nonempty maximal subdomain of D. Remark. The following weak form of the existence theorem is easy to obtain: Every domain G is the maximal domain of existence of a function f E 0(G). Proof. Choose a set A that is locally finite in G arid accumulates at every boundary point of G. By the general product theorem, there exists an f E 0(G) with Z(f) = A. By the identity theorem, there is no holomorphic extension of f to a domain Ô D G.
1. A construction of Goursat. We fix a sequence a l (1 2 , ... in Cx with
E la,' < oc and a sequence 1) 1 . b2, ... of pairwise distinct points
in C. We
denote by A the closure in C of the set Ibl, b2. • • •I• (1)
az,
The series 1(z) = 1.0
=I
converges normally in C\A.
114
5. Iss'sa's Theorem. Domains of Holomorphy
Proof. If K C C\A is compact, the distance d between K and A is positive. Since lz-b,I> d for Z E K, it follows that E a,l(z-bi,)1K < d-1 E aI
lard IW
ig(01
btL1
E
IaI >
v=p+
by
I
—
lard 2 lw - bnl
i„r w \I
11.
Since 19(w)1 remains finite as w approaches b„, the assertion follows. Remark. The statement of the lemma is not obvious once the point 6,, is an accumulation point of other points bk . The growth of the "pole terms" a7,I(z - b„,) bk ) correabout b,, could then be offset by the infinitely many terms ak 1(z sponding to the bk that accumulate at b,,. This phenomenon actually does occur for other series. For instance, every summand of the series -
g(z) =
E 2 11 z 2I' -1 /(1 +z 2v ), v=o
which converges normally in E, has poles on 0E; different summands never have equal poles (so that nothing cancels here), the poles of all the summands are dense in 8E, yet in the limit these poles cancel completely. The limit function, far from having infinitely many singularities on 0E, is just g(z) = 1/(1 - z); this can be seen at once by logarithmic differentiation of the product rir 0 (1 + ze) (cf. Exercise 2 in 1.2.1).
_Historical note. E. Goursat used series of the type E a,/(z - b,) in 1887 to construct functions with natural boundaries [Gout]. A. Pringsheim studied such series intensively; cf. [Pr], pp. 982-990.
§2. Domains of Holomorphy
115
2. Well-distributed boundary sets. First proof of the existence theorem. If b is a boundary point of G, a disc V C G is called a visible disc for b if b E aV; b is called a visible (from G) boundary point of G. In general, domains have boundary points that are not visible. Thus, in squares, the vertices are not visible; in domains with spikes (Figure 8.4, p. 176), there exist boundary curves none of whose points are visible. A set M of visible boundary points of G is called well distributed if the following holds:
(*) Let c E G and let B be a disc about c that intersects aG. Then in the component of B n G containing e there is a visible disc V for some point bEMnB. With the aid of this concept, we obtain a First criterion for domains of holomorphy. If {b 1 ,b2 ,...} is a countable we ll -distributed boundary set, then G is the domain of holontmphy of every function
(z) =
E au /(z — b,),
z E G, where a1, E C x
E 1a1,I < 00.
Proof. f E 0(G) by 1(1) since {b1,b2,...} c aG. Let e E G and let B be the disc of convergence of the Taylor series h of f about c. Suppose that BnOG 0. Then, by (*), in the component W of B nG containing e there is a visible disc V for some point bn E B. Since 1/1147 = fW, Lemma 1 implies that h tends to oc as b„ is approached along the radius of V to b„. Then b„ V B. It follows that B C G.
The following statement is not obvious:
(1)
Any G
C has a countable well-distributed boundary set M.
Proof Let R be countable and dense in G, e.g. R = (Q+ iQ) n G. For each E R, choose b E OG on the boundary of the largest disc V c G about (. The set M of all these visible boundary points b is countable. Now let B be a disc about e E G that intersects OG. If ç E R is chosen close enough to c, then the largest disc V C G about lies, together with in B, and e E V. By the construction of M, V is the visible disc for some point b E M. Since b E OV C B, and since VC.EinG lies in the component of B n G containing e because c E V, (1) is proved.
av,
The construction of the set M by way of (Q + iQ) n G is motivated by the Poincaré-Volterra theorem, which says, among other things, that the Taylor series of f about all the rational complex points yield all possible holomorphic continuations of f. (These "function elements" are dense in the "analytic structure" of f .)
116
5. Iss'sa's Theorem. Domains of Holomorphy
The existence theorem follows immediately from (1) and the criterion above.
Historical note. The proof of the existence theorem given here can be found in Pringsheim's 1932 book [Pr) (pp. 986-988); he credits an oral communication of F. Hartogs. Pringsheim works only with dense sets of visible boundary points. In 1938, J. Besse pointed out drawbacks to such a choice of boundary points (see the following exercises) and eliminated them ([Bes], pp. 303-305). H. Kneser, in his Funktionentheorie, 2nd ed. (pp. 158-159), discusses only the weak form of the existence theorem (see the introduction to this section); his approach is the same as Pringsheim's. Exercises. Prove: a) Well-distributed boundary sets for G are dense in é:)Gr. b) There exist domains for which not every dense set of visible boundary points is well distributed. c) If G is convex, then every dense set of visible boundary points is well distributed. 3.
Discussion of the concept of domains of holomorphy. A function
f E 0(G) is called holomorphically extendible (or continuable) to a boundary point p of G if there exist a neighborhood U of p and a holomorphic function g E 0(U) such that f and g coincide on a component W of UnG, with p E aW; otherwise p is called a singular point of f. In general, the neighborhood U is "large": Thus, for the boundary point 0 of the domain n-itHoo, n] x {i/n}}, there does riot exist any disc B 0 C G '-'- 21 \ Ucc such that 0 lies in the boundary of a component of B n G; every function f := giG with g E 0(C) is of course holomorphically extendible to 0 (with U := C). If there exist discs B c U about p such that B n G is connected, then we can choose U to be such a disc. If G is a convex domain, then for every disc B the region B n G is again convex and thus a domain; the next statement follows. (1) If G is convex and f E 0(G) can be extended holomorphically to p E OG, then there exist a disc B about p and a function g E 0(U) .such
that giB n G = flB n G. By (1), the expression "singular point of f" just introduced agrees for discs with that introduced in 1.5.3.3. We now make precise the notion that a domain of holomorphy is the maximal domain in which a function is holomorphic. Theorem. The following statements about a function f E 0(G) are equiv-
alent: 0 The domain G is the domain of holomorphy of f. . ii) There do not exist a domain 0 G and a function î E 0 (G ) such that the set {z E G rld : f (z) = I" - (z)} has interior points. iii) Every boundary point of G is a singular point of f.
a
§2. Domains of Holomorphy
117
Condition ii) sharpens the maximality property discussed in the introduction, where we required that d D G. To prove this theorem, we need a lemma. Lemma. Let G and d be domains in C, and let W be a component of G n d . Then d n aw C 80. If d 0 G, then d n aw is nonempty (see Figure 5.1).
FIGURE 5.1.
Proof. 1) Let g E dn 8W. Since aw C W C G, it follows that g E G. If g were in G, then, since g E d, it would follow that g E W; but this contradicts g E 8W. Hence g E 0\G = aG. 2) Let d çt G. Then a\W is nonempty, for otherwise the inclusion W c G would imply that W = d; but this would give the contradiction d c G since W C G. Moreover, 6\ W is not open in C since d = W U (6\W), W is open, and d is connected. Let p E 6\ - W but not an interior point of d\W. Then U n W 0 for every neighborhood U of p; that is, p E OW. It follows that pEdn aW. 0 We now prove the equivalence of the statements of the theorem in the non iii) non 0." form "non 0 non ii) non ii). There exists acEG such that the disc of convergence non j) of the Taylor series f of f about c does not lie in G. Since f E 0(d) and 1 1W = 1114/ on the component W of G n d containing c, non ii) follows. non ii) = non iii). Suppose that d 0 G, f E 0(d), and W1 is a component of G n d such that fiWi. = flWi. By the lemma, there exists a point p E d naW1 c aG. We may assume that p is a visible boundary point of W1 (by density; cf. Exercise 2.a)). We choose a disc U C d about p and a visible circle V c W1 for p E am. Then U n V lies in a component W of G n U. That p E ay n aG implies that p E 0W. Set g := flU; then glW = flW since VnU C W1 . Thus p is not a singular boundary point of
f.
118
F. Iss'sa's Theorem. Domains of Holomorphy
non iii) non i). Suppose that p E OG is not singular for f and let U. g, and W be chosen accordingly. Let r be the radius of convergence of the Taylor series of g about p. We choose c E W with le — pl < r/2. The disc of convergence of the Taylor series of y about c then contains the point. p c OC. Since f and g have the same Taylor series about c, G is not the 0 domain of holomorphy of f. Exerrise. If G is convex and is the maximal domain of existence of f E 0(G) in the sense of the introduction, then G is the domain of holomorphy of f.
4. Peripheral sets. Second proof of the existence theorem. A set A that is locally finite in a domain G is called peripheral in G if the following holds:
(*) If G c C is any domain and W is a component of G n point of 6 n OW is an accumulation point of A n W.
6,
then every
With the aid of this concept, we obtain a Second criterion for domains of holomorphy. If the zero set Z(f) of f E 0(G) is peripheral in G, then G is the domain of holomorphy off. Proof. We show that statement iii) of Theorem 3 holds. For contradiction, suppose there exist a point p E 0G, a disc U about p, and a function g E 0(U) such that f IW = gl W on a component W of Gnu with p G Since Z(f) is peripheral in G, pis an accumulation point of Z(f)nW. Since Z(f )nW.Z(g)nW, the identity theorem implies that g O. It follows that f 0, which is impossible since Z(f), as a peripheral set, is discrete in G. O
aw.
It is not obvious that peripheral sets always exist.
(1) If G
C, then there exist peripheral sets A in G.
Proof. Let the set (Q+ iQ)n G be arranged in a sequence (1 , (2 , .... In the largest disc 8' C G about. choose a point a, with d(a,,ac) 1/v. Let A := {a l , a2, ...}. Since every compact set K c G has a positive distance to the boundary d(K, OC), the set A n K is always finite: in other words, A is locally finite in G. Now let G and W be as in (*), and let p Edn OW. Then, for every > 0 with B E (p) c 6, there exists a rational point (k E B E (p) n W with (Ad < .c7. The largest disc Bk about (k that is contained in G now lies in B(p), since p E 8G by Lemma 3. B k n 117 0; hence B k c W since W is a maximal subdomain of G n G. For the point ak E B k corresponding to (k , it now follows that ak e 13,-(p) A n W.* Since e > 0 is arbitrary, (*) holds.
0,
> 0.
These Cassini curves have only ordinary points except in the case a = R, where is a double point (left-hand part of Figure 5.2). The Cassini region D corresponding to (1) has two components if u > R and is connected if a < R. We prove a more precise result.
If a < R, then the Cassini region D is a star-shaped domain with center 0. Proof. In polar coordinates, (1) has the form r 4 - 2a 2 r 2 cos 2. = R4 - (14 . With sin 2 2 - R", it follows that D = {re 'q'E C: g(r2 ) < 0). Since > a I . g has exactly one positive and one negative zero, hence
g(I) := (t a2 (.os2,p) 2 + p2 )
1 denotes a principal part distribution on C. Every function q„ E 0(C\{4}) has a Taylor series about 0, which converges in the disc of radius 141, v > 2 (observe that d1 = 0 is possible). We denote by Po, the kth Taylor polynomial for qi, about 0 (deg pvk < k) and show that these polynomials can serve as convergence-producing summands. Mittag-Leffler's theorem. For every principal part distribution (d,,, qz,),> 1 in C, there exist Mittag-Leffler series in C of the form 00
qi
± E(qv — Ppk,,),
where p i,k„ := ki,th Taylor polynomial of q,, about O.
v=2
Proof. Since the sequence (pi,k)k> 1 converges compactly in B141 (0) to q,,, for every 11 > 2 there exists a ki, E N such that I q,(z) — pvka, I < 2 for all z with I zl < 1141. Since lim 4 = oc, every compact set K in C lies in almost all the discs /31 14i (0). Hence
E i qv
-
Pk,.
1K 5.
v>n
E 2' n
This proves the normal convergence of the series in C\{d1 ,d2, ...}. Since it always holds that py k i, E 0(C), the series in question is a Mittag-Leffler 0 series in C for (d,,, q,)v> 1 . The series 1
oo (
E z v=2
—+
1
z - d,
+
1 d„
+
Z
c/3
zv-2
+
+
,,
d',1, - 1 I
is a Mittag-Leffler series (with q„(z) = (z — 4) -1 and p,_2(z)); it appeared in 3.1.5(1). 4. Consequences. Mittag-Leffler's theorem has important corollaries. Existence theorem. Every principal part distribution on C with support T is the principal part distribution of a function holomorphic in C\T. Theorem on the partial fraction decomposition of meromorphic functions. Every function h that is merornorphic in C can be represented by a series E h,, that converges normally in C, where each summand h„ is rational and has at most one pole in C.
§1. Mittag-Leffler's Theorem for C
129
The existence theorem is clear; the second theorem is proved as follows: By Theorem 3, corresponding to the principal part distribution PD(h) there is a Mittag-Leffler series h in C whose convergence-producing summands are polynomials. Since all the principal parts of h are finite, all the summands of this series are rational functions that have exactly one pole in C. The difference h is an entire function and therefore a normally convergent series of polynomials in C (Taylor series). The Weierstrass product theorem can be obtained from Mittag-Leffler's theorem. We sketch the proof for the case that D is a positive divisor on C satisfying D (0) = 0 and prescribing only first-order zeros. If d 1 , d2 ,... is a sequencecorresponding to D, we consider the principal part distribution (d11 . 1/(z - dv )),, >1 on C. Then -(4) -1 0 (z/d„)k is the WI Taylor polynomial; Mittag-Leffler series for this distribution look like
E k,
with
h,(z) :=
+
z
(
.
ald.
V=1
Every f E 0(C) with f I f = h now has D as divisor. Since k„
--■
1\
+1
(z d
) 1C-1-11 v E N,
,
Weierstrass's assertion that f = H Ek,(z/d,,) follows automatically. Of course, it must still be shown that this product converges. This can be done, for example, by integrating AY fy ; for details, see [FL], pp. 176 177.
5. Canonical Mittag- Leffler series. Examples. In applying Theorem 3, we will choose the numbers k„ as small as possible — as we did for Weierstrass products. If all the k, can be chosen to be equal, the MittagLeffler series q i + (a, - p vk) with the smallest k > 0 is called the canonical series for the principal part distribution (civ , q)„ > , in C. We give four examples. , in > 2, is the canoni1) The Eisenstein series E rn (Z) := Ec-0,3 (z + cal series for the principal part distribution (-u, 1/(z+v) 171 ), E z; convergenceproducing summands are unnecessary here. We recall the explicit formulas (1.11.2.3) 00
1
7r3
(z + 14 2 '
sin2 7rz
cot 7T Z sin2 7rz
=
2_, , (z + v) 3
2) The cotangent series DO
r cot
rz
El
(z) = - + E z
( 1
1 z+v y)
130
6. Functions with Prescribed Principal Parts
is the canonical series for the principal part distribution (—v. 1/(z + v))vEz; here k = U.
3) The series
P(z)
1
1 z
1'W
EQ
1) (z „
(cf. 2.2.3),
if we disregard the terni --y, is the canonical series for the principal part distribution (—i), —1/(z + v)), >0 . Again k = 0. 1) The Eisenstein-Weierstrass series
p(z) :=
1
1 tz+ , ) ,
1)
k
is the can °meal series for the principal part distribution (—w, 1/(z+4.0) 2 ), E 0. where SI denotes a lattice in C; cf. 3.2.4. Here also, k = 0. These examples were well known before Nlittag-Leffler proved his theorem. The t heorem shows that the same construction principle underlies all four examples.
6. On the history of Mittag-Leffler's theorem for C. hi connection vit Ii t he research of Weierstrass, which was published in 1876 ([Wei], pp. 77 124), Mittag - Leffler published his theorem in 1876 77 for the case that all t he principal parts are finite, in Swedish, in the Reports of the Royal Swedish Academy of Sciences, Stockholm (cf. [ML], p. 20 and p. 21). In his 1880 note [Wei], pp. 189 199, Weierstrass simplified the proof considerably by introducing the Taylor polynomials as convergence-producing summands: in this work, Weierstrass also drew attention to the theorem proved by Nlittag-Leffler in 1877 on the partial fraction decomposition (cf. pp. 194-195): Es ldsst sich also jede eindelitzge analyttsche Function f(i), fdr die im Endlichen keine wesentliche smguldre Stelle existirt, ais von rationalen Functionen der Verdnderlichen einSU17 dergestalt ausdrücken, dass jede dieser Fun ctionen zrn Endlichen hiichstens eine Unendlich,keits-Stelle hat. (Thus every singlevalued analytic function f (x) for which no essential singularity exists in the finite plane can be expressed as a sum of rational functions of the variable x in such a way that each of these tunet ions is infinite at no more than one point in the finite plane.) The derivation of the Weierstrass product theorem for C from MittagLefller's theorem for C by integrating logarithmic derivatives was communicated to Nlittag-Leffler by Hermite in a letter in 1880 ([Hen, especially
§2. Mittag-Leffler's Theorem for Arbitrary Regions
131
pp. 48-52). This method of proof found its way into textbooks at the beginning of this century. A.Pringsheim expressed his indignation with this approach in 1915 ([P], p. 388): "Wenn nun aber einige Lehrbiicher sich so weit von der Weierstraf3chen Methode entfernen, dal3 sie den fraglichen Satz ais Folgerung (!) a. us dem Mittag-Leifierschen Satze durci logarithmische Integration herleiten (und zwar dieses Verfahren nicht etwa nur in Form einer gelegentlichen, ja sehr nahe liegenden Bemerkung, sondern ais einzigen und mafigebenden Beweis mitteilen), so diirfte diese Art, die Dinge auf dem Kopf zu stellen, wohl von niemandem gebilligt werden, der in der NIathernatik etwas anderes sieht, ais eine regellose Anhdufung mathematischer Resultate." (But even if some textbooks now deviate so far from Weierstrass's methods that, using logarithmic integration, they derive the theorem in question as a consequence (!) of Mittag-Leffier's theorem (and in fact present this method not just in the form of an incidental, indeed quite obvious, remark, but as the only and standard proof), this topsy-turvy way of doing things should not be sanctioned by anyone who sees mathematics as something other than a disordered heap of mathematical results.) Interesting details of the history of ideas of Mittag-Leffler's theorem can be found in Y. Domar's article [D].
Mittag-Leffier's Theorem for Arbitrary Regions As always, D denotes a region in C. Our goal is to construct Mittag-Leifier series in D for every principal part distribution in D. We consider only principal part distributions (d„, qi,) with infinite supports. NIittag-Leffler series for special principal part distributions are given in Subsection L The general case is handled in Subsection 2.
1. Special principal part distributions. We begin by proving the following: Let q(z) E 0(C\01) be a principal part at d E C, and let c E C\{d}. Then, in the annulus {z E C : 1.2 — c> d— cl} about c, q has a Laurent expansion of the form Ei7ap (z — c)P. Proof. For all p> Id — c, the coefficients a p of the Laurent series of q in the given annulus satisfy the Cauchy inequalities
M(P) := max{19(z)1 z E
E Z.
M (p) = 0 since lim, q(z) = 0 (cf. 1.1); hence a p = 0 for all > O.
We call gk (z) := E-k E 0(C\IcI) the kth Laurent term t1=-1 aP (z — of q about c. We will see that in special situations such Laurent terms can serve as convergence-producing summands for Mittag- Leffler series.
132
6. Functions with Prescribed Principal Parts
(4, q„),>1 be a principal part
distribution on D with support T. As in 4.1.2, let T' := --TAT denote the set of all the accumulation points of T in C; this is closed in C. Then (dv, qv)v>i can be interpreted in a natural way as a principal part distribution on the region C\T' D D. Let
Proposition. Let a sequence (c,), >1 in T' be given with lim1d,, — c v l = O. Let g,,k denote the kth Laurent term of q„ about c„. Then there exist (many) sequences (k„)„>1 of natural numbers such that Evce=i (qt, — gvic,) is a Mittag-Leffler series for (q,, d),,>1 in C\T' Proof. Since the sequence (g 11 k)k>0 converges uniformly to q in {z E C : — c > 21 4 — cy l } , for every u > 1 there exists a k, E N such that
lq,(z) — g,k ,,(z)1 < 2' for all z E C with z
— Cz,
I
>
214
Now let K be a compact set in C\11 . Since d(KT) > 0 and limn there exists an 71(K) such that, for all u >
K c {z E C : lz — cv l 2Id, — c,1 and hence
E
q1, — g11kjK
1 , with Z := v(d,) and (7,, := q,, o v -1 — q,,(a), is the principal - „ — 0 v ; it part distribution transported by y to C\v(r). By (*), q,, — g,, = ( 4 follows that E (q,, — g 1,) is a Mittag-Leffler series in C\T' , as desired. Details are left to the reader.
A third short proof, using Runge theory, is given in 13.1.1.
3. Consequences. We first note the following: Existence theorem. Every principal part distribution on an arbitrary region D c C with support T is the principal part distribution of a function holomorphic in D\T Every distribution of finite principal parts on D is the principal part distribution of a function m,eromorphic in D. Theorem on the partial fraction decomposition of meromorphic functions. Every function meromorphic in D c C can be represented by a h, of meromorphic functions partial fraction series; that is, by a series on D that converges normally in D, where each function h,, has at most one pole in D.
E
By combining the general product theorem 4.1.3 with the general MittagLeffler theorem, we obtain
Mittag-Leffier's osculation theorem. Let T be locally finite in D and let every point d E T be assigned a series vd(z) = Endo° ad,(z — d)" that converges normally in CV, where nd E N. Then there exists a function h, holomorphic in D\T , whose Laurent expansion about d has the function v d as "section"; that is, od(h vd) > rid for all d E T. Proof. Let f G 0(D) be chosen such that rid < od( f) < oc for all d E T. We consider the principal part distribution (d, qd)d E T on D with support T. where qd is the principal part of vdl f at d. Let g e 0(D\T) be a solution of this distribution. Then h := f • g is a function with the desired properties. To begin with, it is clear that h is holomorphic in D\T. The functions Pd := g
—
qd
and
rd := (vdl f) — qd
are holomorphic in a neighborhood of d E T. Since the equation h = f (pd + qd) = f • (vdI f + pd — rd ) = vd + f • (pd — rd)
134
6. Functions with Prescribed Principal Parts
obviously holds in a neighborhood of d E T and since pd - rd is holomorphic in a neighborhood of d, it follows that od (h, - vd) > nd for all d E T. A special case of the osculation theorem is the Interpolation theorem for holomorphic functions. Let T be locally finite in D let a polynomml pd(z) =E (7)id ad„(z - d)" be assigned to every point d E T. Then there exists a function f holomorphic in D, whose Taylor series about d begins with the polynomial pd, d E T. For the special case D = C, this theorem means that there always exist
entire functions that have arbitrarily prescribed values on a sequence d1 , d2 .... without accumulation points in C. This statement generalizes the well-known Lagrange interpolation theorem, which says that given n distinct points d i , , d„ and n arbitrary numbers w 1 , , /E n , there always 1 with p(4) = w,,, exists (exactly) one polynomial p(z) of degree < n 1 < y < n, namely -
p(z) =
E w,, H (z _ d„ ) ,(d, _ do ). 1,1
1,41,
(This is the Lagrange interpolation formula.) Exercise. Let (a,,),,›0 be a sequence of pairwise distinct complex numbers with a () = 0 and lim a„ oo. Assume that f E 0(C) has no zeros in C\fao , and that ou „ ( f) = 1 for all I/. Then for every sequence (b,),>0. b,, E C, there exists a sequence (n,,),>o, with n,, E N, such that the series bof (z)
z f ' (0)
v. 1
(z) f ' (a ,,)(z ay )
az,„
converges normally in C to a function F E 0(C). Moreover, F(a,„) = b,, for v > O.
4. On the history of Mittag-Leffler's general theorem. Theorem 2 was stated in 1884 by Nlittag-Leffler ([ML], p. 8). With Weierstrass's encouragement, he had worked on these questions since 1876 and had published several papers about them in Swedish and French. Y. Domar writes ([D], p. 10): "The extensive paper [ML] is the final summing-up of Mittag-Leffler's theory .... The paper is rather circumstantial. with much repetition in the argumentation, and when reading it one is annoyed that Mittag-Le ffl er is so parsimonious with credits to other researchers in the field, Schering, Schwarz, Picard, Guichardi yes. even Weierstrass. But as a whole, the exposition is impressive, showing Mittag-Leffler's mastering of the subject." In [ML] the osculation theorem is also treated for the case of finite principal parts (cf. p. 43 and pp. 53-54); our elegant proof can be found in the 1930 paper of H. Caftan ([Cad, pp. 114-131). The paper [ML], in which
§3*. Ideal Theory in Rings of Holomorphic Functions
135
ideas of G. Cantor already appear, contributed to Mittag-Leffler's reputation in the mathematical world (see his short biography on p. 323). The results immediately exerted a great influence; C. Runge wrote his groundbreaking paper on approximation theory under the impact of Mittag-Leffler's work (cf. 13.1.4 and Runge's short biography on p. 323). H. Behnke and K. Stein showed in 1948, with methods of the function theory of several variables, that Theorem 2 and hence its corollaries in 3 hold verbatim if one admits arbitrary noncompact Riemann surfaces instead of regions in C (cf. [BSt2], Satz 1, p. 156); the osculation theorem can be found at the end of this paper as Hilfsatz C. 5. Glimpses of several variables. In his paper [Col. already mentioned in 4.2.4, Cousin extended Mittag-Leffler's theorem to polydomains in C". As in the case of the product theorem, difficulties arise even in formulating the problem: the concept of the principal part distribution must be understood in a different way. since the poles of meromorphic functions are no longer isolated, but - like their zeros form real (2n - 1)-dimensional surfaces. Furthermore, the various surfaces can intersect each other; this occurs, for instance, for the function te/z Surprisingly, it turns out that despite these complications the situation is nicer than when the Weierstrass product theorem is extended to higher dimensions; cf. 4.2.4. No topological obstructions appear! H. Cartan first observed, in 1934, that a domain in C2 for which Mittag-Lefiler's theorem holds must be a domain of holomorphy ([Ga i ], p. 472); a proof of this was given in 1937 by H. Behnke and K. Stein ([BStd, pp. 183-184). In the same year, K. Oka [0 1 ] succeeded in proving that Mittag-Leffier's theorem holds for all domains of holomorphy in C. In 1953. H. Cartan and .J-P. Serre finally proved again using sheaf theory and cohomological methods — that, in any Stein space there always exist meromorphic functions corresponding to prescribed principal part distributions (ra21, p. 679). The reader can find precise forms of these statements in [GR, especially pp. 140 142: the monograph [BT] contains further historical details.
§3*.
Ideal Theory in Rings of Holomorphic Functions Der WeierstraBsche Produktsatz lehrt mis, daii in den Bereichen der z-ebene aile lendlich erzeugten] Ideate Hauptideale sind. (The Weierstrafi product theorem teaches us that in the regions of the z-plane all [finitely generated] ideals are principal ideals )
— II. Behnke, 1940
136
6. Functions with Prescribed Principal Parts
0 of a commutative ring R with unit element 1 Recall that a subset a is called an ideal in R if ra + sb E a for all a, b E a and all r, s E R. If M 0 is any subset of R, then the set of all finite linear combinations r,f,, f E M, is an ideal in R with generating set M. Ideals a that have a finite generating set {fi, , f„) are called finitely generated: the suggestive notation a = Rf i +••• + Rfn is used in this case. Ideals of the form Rf are called principal ideals. A ring R is called Noetherzan if every ideal in R is finitely generated, and a principal ideal ring if every ideal is a principal ideal. One of the goals of this section is to show that in the ring 0(G) of all functions holomorphic in a domain G C C, every finitely generated ideal is a principal ideal (Subsection 3). Our tools are a lemma of Wedderburn (Subsection 2) and Theorem 4.2.1, on the existence of the gcd; thus MittagLeffler series and Weierstrass products form the basis for the ideal theory of 0(G).
E
1. Ideals in 0(G) that are not finitely generated. Let A be an infinite locally finite set in G. The set
a :=
E 0(G) : f vanishes almost everywhere on A}
is an ideal in 0(G). If f„ are arbitrary functions in a, the set of their common zeros again consists of almost all the points of A. By the existence theorem 4.1.5, for every point a e A there exists an f G a with f (a) 0 0; hence the ideal a is not finitely generated. We have proved the following result.
No ring 0(G) is Noetherian; in particular, 0(G) is never a principal ideal ring. Exercise. Let G := C and let a denote the ideal in 0(G) generated by the functions
sin rz fJ (z —
E 0(C), n E N.
= -11
Is a finitely generated?
Because of what we have just proved, the ideal theory of the rings 0(G) is necessarily more complicated than the ideal theory of Z, Z[i], or the polynomial rings C[Xi , , Kn.] in finitely many indeterminates. Nonetheless, we will see that 0(G) has an interesting ideal-theoretic structure. Our starting point is
2. Wedderburn's lemma (representation of 1). Let u, .1; E 0(G) be relatively prime. Then they satisfy an equation au + by = 1
with functions a,b E 0(G).
§3*. Ideal Theory in Rings of Holomorphic Functions
137
0. Since 1 = gcd{u, v } implies that Proof. We may assume that ut' Z(u) n z(v) = 0, the pole set of 1/uv is the disjoint union of the pole sets of 1/u and 1/v. By rearranging a (normally convergent) partial fraction series for 1/ut' (using Theorem 1.4), we thus obtain l/uv = a 1 +1)1, a 1 ,h1 E M(G), where a l has poles (of order —oe (v)) only at the points e of Z(v) and bi has poles (of order —oc (u)) only at the points c of Z(u). Then a := and b := ubi are holomorphic in G, and it follows that an + by = I. Historical note. The lemma just proved was published in 1915 by J. H. M. Wedderburn; we have reproduced his elegant proof, which is almost unknown in the literature ([Wed], p. 329). The trick of splitting the pole set of a meromorphic function h into two disjoint sets PI and P2, and writing a Mittag- Leffler series for h as the sum h 1 + h2 of two such series, with P(h i ) = P1 and P(h2) = P2, had already been used in a different context by A. Hurwitz in 1897 ([Hul, p. 457). Wedderburn's lemma reappears implicitly in a 1940 paper of O. Helmer for G = C ([Hell, pp. 351-352); he considers entire functions with coefficients in a prescribed fixed subfield of C. Helmer is not familar with Wedderburn's work. We give a second proof of Wedderburn's lemma, using Mittag-Lefiler's osculation theorem, which even gives a sharpened version: If u, y E 0(G) are relatively prime, then there exist functions a, b E 0(G) such that au + by = 1, a has no zeros in G. Proof. If v 0, set a := 1/u, b := 0. Suppose r O. It suffices to show that there exist functions A, h E 0( 0 ) with u — Ay = e h , since a := e. := —Ac -h will then give the desired result. Since Z(u)n z(v) =0, for every c E Z(v) there exist a disc U, C G about c and a function fc E 0(U„) such that ulU, = . Since Z(v) is locally finite in G, by the osculation theorem 2.3 there exists h E 0(G) such that oc (h — fc ) > oc (v). c E Z(v). Noting that Meg — 1) = oc (q) whenever q vanishes at c, we see that
oc (u — e h ) = oc (eh — e h )
= oc (e fe" — 1) =
(Of, — h),
c E Z(v).
Hence A := (u — e h )ly E 0(G). The preceding proof was sketched in 1978 by L. A. Rubel, who, however, uses Wedderburn's lemma [Rub]. In general, it is impossible to choose both the functions a and b to be nonvanishing; for instance, when G = C and u = I. v = z ([Rub], p. 505). Exercise. Prove that for every g E A/1(G) there exists an f E 0(G) such that f(z) g(z) for all z E G. Hint. Start with a representation g =
12 with relatively prime f 1 , 12 E 0(G).
138
6. Functions with Prescribed Principal Parts
3. Linear representation of the gcd. Principal ideal theorem. In 4.2.1 we saw that every nonempty set, in 0(G) has a gcd. Wedderburn's lemma makes it possible, in important cases, to represent the gcd additively, Proposition. If f E 0(G) is a gcd of the finitely many functions f E 0(G), then there exist functions ..... a, E 0(G) such that f = al fi +
a2f2 +
In
+ an f n•
Proof (by induction on n). Let f 0. The case n = 1 is clear. Let n> 1 and let f := gcd{f2 , , f71 } . By the induction hypothesis, f = a2f2+ • • with a2,...,an E 0(G). Since f = gcdtf, n by 4.2.1, it follows that u := ft/f, y := flf E 0(G) are relatively prime. Thus by Wedderburn there exist a, b E 0(G) such that 1 = au+ by. Hence f = aif, + • • + (ink with a l := a, au := biz, for v > 2. One important consequence of the proposition and the existence of the gcd is the Principal ideal theorem. Every finitely generated ideal a in 0(G) is a principal ideal: If a is generated by fn, then a = 0(G)f,, where f = gedffi, ••• ,
•
Proof. By the proposition, 0(G) f C a. Since f divides all the functions fi, • • • , fn, we have f l ,...,f„ E 0(G) f ; hence a c 0(G) f The proposition and the principal ideal theorem, together with their proofs, are valid for any integral domain R with gcd in which the statement of Wedderburn's lemma is true. --- Integral domains in which any finite collection If', • . • , always has a gcd are sometimes called pseudo-Bézout domains; they are called Bézout domains if this gcd is moreover a linear combination of the f„ (cf. [Bo], pp. 550-551, Exercises 20 and 21). 0(G) is thus a Bézout domain.
Exercise. Let A denote the set {sin 2 - nz : n E N} in 0(C). Is the ideal generated by A in 0(C) a principal ideal?
4. Nonvanishing ideals. A point c E G is called a zero of an ideal a in 0(G) if f(c) = 0 for all f E a, i.e. if a c 0(G) • (z — c). We call a nonvanishing if a has no zeros in G. An ideal a in 0(G) is called closed if a contains the limit function of every sequence f„ E a that converges compactly in G. Proposition. If a is a closed nonvanishing ideal in 0(G), then a = 0(G).
For the proof, we need a
§3*. Ideal Theory in Rings of Holomorphic Functions
139
Reduction rule. Let a be an ideal in 0(G) for which the point C E G is not a zero. Let f, g E 0(G) be such that fg E a and, if f vanishes anywhere ni G, it does so only at c. Then g E a. Proof. Choose h E. a with h(c)
O. Let n, := (Of). If n > 1, then
1 pt(z) - h(c) g =
fg
fg E a. -• iii
Applying this n times gives [f/(z - c ) ] g E a. Since f 1(z - c)" is invertible in 0(G), it follows that g E a. The proof of the proposition now goes as follows: Let f E a, f O. Let 11 f„ be a factorization of f, where L, E 0(G) has exactly one zero e„ in G. Then the sequence f„:= f,, E 0(G) converges compactly in G to
1 (cf. 1.2.2). We have = f„f 1+1 . Since fo f E a and f„ has no zeros in G\fe n l. it follows (inductively) by the reduction rule that f„ E a for all n > O. Since a is closed, it follows that 1 E a; hence a = 0(G). The hypothesis that a is closed is essential for the validity of the proposition: the ideals given in Subsection 1 are nonvanishing but not finitely generated and hence also not closed. Exercise. Let a 0(G) be a nonvanishing ideal in O(G). Prove that every function f E a has infinitely many zeros in G.
5. Main theorem of the ideal theory of 0(G). The following statements about an ideal a C 0(G) are equivalent. i) a is finitely generated. ii) a is a principal ideal. iii) a is closed. Proof i) ii): by the principal ideal theorem; ii) i): ii) iii): Let a = 0(G)f O. and let gn = a„ f c a be a sequence that. converges compactly to g E 0(G). Then a,, = g„ If converges compactly in G\Z ( f ) and therefore, by the sharpened version of the Weierstrass convergence theorem, in G, to a function a E 0(G) (cf. 1.8.5.4). It follows that g = of E a. iii) ii): By Theorem 4.2.1, a has a greatest common divisor f in 0(C). Then a' := f -l a is a nonvanishing ideal in 0(G). Since a' is closed if a is . Proposition 4 implies that a' = 0(G). Hence a = 0(G) f Corollary. The following statements about an ideal ni alent.
c 0(G) are equiv-
140
6. Functions with Prescribed Principal Parts
i) m is closed and a maximal ideal in 0(G).' ii) There exists a point c E G such that m=ff E 0(G) : f (c) = 01 iii) There exists a character 0(G) -4 C with kernel m. Details are left to the reader, who should also convince himself that there exist uncountably many maximal ideals in 0(G) that are not closed. One might think that for maximal ideals m in 0(G) that are not closed, the residue class fields 0(G)/m would be complicated. In 1951. however, M. Henriksen used transfinite methods to prove (for G = C) that 0(G)/m, as a field, is always isomorphic to C ([Hen], p. 183); these isomorphisms are extremely pathological. All the results obtained in this section remain true if we admit noncompact Riemann surfaces instead of domains in C. The theorems of Weierstrass and Mittag-Leffler are at our disposal in this situation (cf. 4.2.4 and 2.4): hence so are Wedderburn's lemma and the existence of a gccl for arbitrary noncompact Riemann surfaces, and we can argue just as for domains.
6. On the history of the ideal theory of holomorphic functions. The ideal theory of the ring 0(G) was not developed until relatively late in the twentieth century. Mathematicians of the nineteenth and early twentieth centuries had no interest in it. As early as 1871, R. Dedekind had completely mastered the ideal theory of the ring of algebraic integers (cf. his famous supplement to the 2nd edition of Dirichlet's Vorlesungen über Zahlentheorie, and also Dedekind's Gesammelte Matheinatische Werke, vol. 3, pp. 396-407). But even a great algebraist like Wedderburn, who was certainly familiar with Dedekind's theory, and who in 1912, with his lemma, already held the key to ideal theory in arbitrary domains G c C, said nothing about ideal theory. His goal - as the very title of his work [Wed] indicates — was to obtain normal forms for holomorphic matrices (cf. also [N], p. 139 if). It was not until Ernmy- Noether who, incidentally, is credited with the statement: "Es steht alles schon bei Dedekind" (It's all in Dedekind already) — that the ideal theory of rings that are not Dedekind was also considered. The ideal theory of the ring 0(C) was first considered in 1940 by O. Helmer. Helmer admits subfields of C. He first proves Hurwitz's observation (cf. 3.2.5*), without referring to Hurwitz ([Hel], p. 346). Helmer's main result is that finitely generated ideals are principal ideals ([Hel], p. 351); to this end, he proves Wedderburn's lemma -- compared to Wedderburn, in a rather complicated way. Among function theorists in several variables, the principal ideal theorem 3 was already folklore by around 1940; the epigraph 'An ideal m R of a ring R is called maximal if R is the only ideal in R which properly contains m. With the aid of Zorn's lemma, it can be shown that every ideal a R is contained in a maximal ideal.
§3 11 . Ideal Theory in Rings of Holomorphiic Functions
141
of this section is in Behnke's report (Fortschr. Moth. 66, p. 385 (1940)) on Carton's work ([Ca2), pp. 539-564); cf. also Subsection 7. A paper by O. F. G. Schilling, in which our Proposition 4 for the case G C occurs as Lemma 1, appeared in 194G (1S(.1 p. 949). Further papers then appeared in rapid succession; in these, arbitrary domains in C and finally arbitrary noneompa.et Riernann surfaces were admitted. The approach at that time was somewhat different: the principal ideal theorem was proved first, and everythinf.ç else was derived from it. Of these numerous publications, we have included in our bibliography only the 1979 paper [A] of N. L. Ailing, which gives a good overview of the status quo. All these papers have a strung algebraic flavor, and offer little to the reader primarily interested in function theory. 7, Glimpses of several variables. The ideal thtawy of holomorphic functions of several variables — ixcontrast to that of one variable — has long been a focal point of research, and has been an essential factor in shaping the higherdimensional theory. The focal theory was already developed in 1931 W. Rückert, a student of W. Krill], then proved in his- paper MC] 1 published only in 1933 and now considered a classic, that the ring of all convergent power series in n variables., 1 0 converges at a.; b) the sequence (ftn)rt>o, I>l, is a subsequence of (A- 3.,n)n>o•
We argue inductively. Given the sequences („fkn)n>o, k < I, choose a subsequence (fi)>o of the sequence (fi_ i , n )„>0 which converges at at. Then a) and b) are satisfied for all sequences (fk i,),>0, k < 1. From the sequences f/o, fa, f12, • • . , we now construct the diagonal sequence go, gi, g2,... , with gn := fnn., n E N. It converges at every point a, E A since, by b), from the term gr„ on it is a subsequence of the sequence fm, fna, fm2, , which converges at a, by a). The set A must be countable for (*) to hold. The subsequence gn obtained above cannot be expected to converge pointwise everywhere in D. With suitable hypotheses on the sequence fo, fi, f2, , h owever, this does occur. We can even obtain compact convergence, as will now be shown. 1. Montel's theorem for sequences. A family F c 0(D) is called bounded in a subset A c D if there exists a real number M > 0 such that M for all f G .7* (equivalently, supfe7supzE A If(z)I < oc). If IA The family .7' is called locally bounded in D if every point z E D has a neighborhood U c D such that .F is bounded in U: this occurs if and only if the family .F is bounded on every compact set in D. In particular, a family .F c 0(B) in a disc B = Br (c), r > 0, is locally bounded in B if and only if it is bounded in every disc Bp (c), p < r.
§1. Montel's Theorem
149
Bounded families are locally bounded; the converse is not true, as is shown, for instance, by the family Inz" E 0(E), n E NI. A sequence Jo, fi f2 , ... of functions f, E 0(D) is called (locally) bounded in D if the family { fo , fi. f 2 ,...} is (locally) bounded in D. The following theorem now holds.
Montel's theorem (for sequences). Every sequence fo , fi. f 2 ,... of holoinorph,tc functions in D that is locally bounded in D has a subsequence that converges compactly in D. Warning. The assertion of the theorem is false for sequences of real-analytic functions: the sequence sin TIX, n E N, which is bounded in R, does not even have pozntwise convergent subsequences. In fact: The set Ix e R : limi„ sin no- exists} has Lebesgue measure zero for every sequence n i < n2 < • - • in N.
This statement. is consistent with the Arzelà-Ascoli theorem since the sequence sin na' is not locally equicontinuous; cf. 2.2. Montel's theorem will be proved in the next. subsection; we use the following lemma. Lemma. Let .F C 0(D) be a locally bounded family in D. Then for every point e E D and every E > 0 there exists a disc B C D about c such that If (w) — f (z)l
0 so small that. B2,.(c) C D. We set /73 := B,..(c) and := B2r (e). It follows from the Cauchy integral formula f (w) — f ( z) =
1 27ri 2,1.! — Z
27ri
f(()
J
1
1
[c — 1 1' f(()
n'
(C T-1 , )(( —
d(
—Z
z)
d(
and the standard estimate — since 1(( — w)(‘ — z)1 > r 2 for all w, z E — that. E If (w)- f(z)I <w - zi —2r f IB , for all w, z E h and all f E F. f G 1. } < oc; Since .F is locally bounded, M := (2/r) • sup{1f1 B , we may assume that M > O. It now suffices to set B := B(c) with b := minfe/(2/11),r1.
The lemma says that locally hounded families are locally equicontinuous; cf.
150
7. The Theorems of Montel and Vitali
2. Proof of Montel's theorem. We choose a countable dense set A C D, for instance the set of all rational complex numbers in D. By (*) of the introduction, there exists a subsequence gn, of the sequence fr, that converges pointwise on A. We claim that the sequence gn converges compactly in D. To prove this, we need only prove that it converges continuously in D,2 i.e. that lim g„(z„) exists for every sequence z n E D with z„ = f E D. Let E > 0 be given. By Lemma 1, there exists a disc B C D about z* such that (w) — g, 1 (z) 1 < E for all n if w, z E B. Since A is dense in D, there exists a point aeAn B. Since lim.zn = e, there exists an n1 E N such that z„ E B for all n > n1. The inequality
— g„
(.z77,) — 9,11(0 + 1.9m, (a) —
+ ign (z.) — 971(0
always holds; hence Ig„(z) — 9,, (z n )I 2E + I g ni (a) g (a)1 for all m, n > n 1 . Since lim gn (a) exists, there is an n2 such that Igyn(a) — g n (a)1 < E for all m, n > n2 . We have proved that Igrn (zyn ) — gn(z,)1 < 3e for all in, n > max(n i , n2); thus the sequence gn (zn ) is a Cauchy sequence and therefore convergent. —
We comment on the history of IVIonters theorem in 2.3. The theorem is often used in the following form:
3. Montel's convergence criterion. Let fo, fi, 12... be a sequence of functions f„ E 0(D) that is locally bounded in D. If every subsequence of the sequence f converges compactly in D converges to f E 0(D), then f„ converges compactly in D to f. Proof If not, there would exist a compact set K C D such that Ifn — fi did not converge to zero. There would then exist an E > 0 and a subsequence g3 of the sequence fn such that ig3 — f1K > E for all j. Since the sequence g3 would also be locally bounded, by Theorem 1 it would have a subsequence h.k converging compactly in D. But Ihk fIK > e for all k, so f would not be the limit of this sequence. But this is a contradiction. 0 —
As a first application, we prove
4. Vitali's theorem. Let G be a domain in C, and let fo , fi 12,... be a 2 A sequence 14, E C(D) converges continuously to h : D ---■ C if lim h(z) = h(z`) for every sequence z„ E D with lim z„ = f E D. Continuous convergence of the sequence ht, in D is equivalent, to compact convergence in D (the reader may either prove this or refer to I.3.1.5*). If a sequence hi., EC(D) is such that, for every sequence z„ E D with lirn zr, = z* E D, the sequence of complex numbers h,, (z,,) is a Cauchy sequence, then obviously the sequence h,, converges continuously in D to the limit function defined by h(z) := lim h„(z), z E D.
§1. Montel's Theorem
151
sequence of functions J ., E 0(G) that is locally bounded in G. Suppose that
the set A :=
E G : lirn f„(w) exists in CI
has at least one accumulation point in G. Then the sequence fo , fi, converges compactly in G.
h, • • •
Proof. Because of Montel's criterion 3, it suffices to prove that all the compactly convergent subsequences of the sequence fi, have the sanie limit. But this is clear by the identity theorem, for any two such limits must agree on the set A, which has accumulation points in G.
Imitating this proof gives Blaschke's convergence theorem. Let f„ E 0(E) be a sequence bounded in E. Suppose there exists a countable set A -= 02, E, 'with E(1- lay ') = oc. such that limp f„ (aj ) exists for every point a i E A. Then the sequence f r, converges compactly in E. Proof. Let f G 0(E) be limits of two compactly convergent subsequences of the sequence f„; then flA = flA. Now f, f are both bounded in E. By the identity theorem 4.3.2, it follows that f = f. Montel's convergence criterion yields the assertion.
For the history of Vitali's and Blaschke's theorems, see 3.3. We will encounter other compelling applications of Montel's heorem in the proof of the Riemann mapping theorem in 8.2.4 and t he theory of automorphisms of bounded domains in Chapter 9. 5*. Pointwise convergent sequences of holomorphic functions. In the theorems of Montel and Vitali, can the hypothesis of local boundedness be dropped if the sequence is assumed to converge at all points? The answer is negative: in 12.3.1. we will construct sequences of holomorphic fund ions that converge pointwise but not compactly and whose limit functions are not holomorphic. Such limit functions must, however, he holomorphic aim ost everywhere, as we now prove. Theorem (Osgood, 1901, [0], p. 33). Let fo , f, f 2 ,... be a sequence of functions holomorphic in D that converges pointwise in D to a function f. Then. this sequence converges compactly on a dense subset D' of D: ni particular, f is holoinorphic in D'.
We base the proof on the following lemma. Lemma. Let .7- be a family of continuous functions f : D C that is every set { f (z) : f E T } , z E D. is bounded pointwzse bounded in D
152
7. The Theorems of Montel and Vitali
in C). Then there exists a nonempty subregion D' of D such that the restriction of the family to , If ID' : f E is locally bounded in D'.
Proof (by contradiction). Suppose the assertion is false. We recursively construct a sequence go , in T and a descending sequence K0 D K1 D • • • of compact discs K„ C D such that 19„(z)1 > ri for all z E Kn . Let go 0 be in .F, and let K0 C D be a compact disc such that 0 g0 (K0 ). Suppose that g 1 and K r,_ 1 have already been constructed. By hypothesis, there exist a g.„ E .7. and a point z„ in the interior of K,,_ 1 such that 1g7, (z„)I > n. Since g,, is continuous, there exists a compact disc K, C Kn _i about zrz such t hat 1g „(z„)1 > n for all z E K„. The set nK„ is nonempty. For each of its points z*, we have lg„(z*)1> n for all n, contradicting the pointwise boundedness of T. Osgood's theorem now follows by applying the lemma to the family and all subregions of D: We obtain a dense subregion D' fi, f2. such that the sequence is locally bounded. By Vitali, it then converges compactly in every connected component of D': the limit function is holomorphic in D' by Weierstrass.
§2.
Normal Families
Montel's accumulation-point principle carries over immediately from sequences to families. In the classical literature, the concept of a "normal family" took shape in this setting; it is still widely used today. 1. Montel's theorem for normal families. A family .F C 0(D) is called normal in D if every sequence of functions in .F has a subsequence that converges compactly in D. We immediately make an obvious
Remark. Any family .F D.
C
0(D) that is normal in D is locally bounded in
Proof. It must be shown that the number sup{ :f En is finite for every compact set K. If this failed to hold for some compact set L C D, then there would exist a sequence f„ E .F with limn— oc lf„1= oc. This sequence f„ would have no subsequence converging compactly in D, since for its limit f E 0(D) we would have If1L > IfnILIf fnIL. Contradiction!O The converse of the remark above is the general form of Mantel's theorem. Every family .F C 0(.130 ) that is locally bounded in D is normal in D.
This statement follows immediately from Montel's theorem 1.1 for seqt iences.
§2. Normal Families
153
By what has been shown, the expressions "normal family" and "locally bounded family" are equivalent. A colleague reports that in the early forties, after proving this equivalence in his lectures, he received an inquiry from a higher-level administrative department in Berlin asking whether he would also consider the latest findings of racial theory.
Examples of normal familieS. 1) The family of all holomorphic mappings from D to a (given fixed) bounded region D' is normal in D. 2) For every M > 0, the family
.FA,f := If =
ay e : la,' < M for all v E N}
is normal in IE: For every r E (0,1) and every f E F,ç, we have 1 f(z)1 < M(1 - 0 -1 for all z E Br (0), whence TM is locally bounded in E. 3) If .F is a normal family in D, then every family {f (k) is also normal in D.
:f
E
k E N,
Proof. Since .F is locally bounded in D, for every disc B = B27.(c) with C D there exists an M > 0 such that IfIg < M for all f E F. Using the Cauchy estimates for derivatives and setting ji := Br (c) , we have (cf.
1.8.3.1)
I f (k) 1/3- < 2(Mir k ) • k!
for all f E ,F and all k E N.
The family {P k) : f E .F } is thus bounded in /3 for fixed k E N. Hence it is locally bounded and therefore normal in D.
Another example of a normal family can be found in 4* • Remark. In the literature, the concept of a normal family is often defined more generally than here: The subsequences are also allowed to converge compactly to oc. This formulation is especially useful if one wants to include meromorphic functions as well.
2. Discussion of Montel's theorem. Montel's theorem — in contrast to that of Vitali - is not, strictly speaking, a theorem of function theory, as it can easily be subsumed under a classical theorem of real analysis. We need a new concept. A family F of functions f : D C is called equicontinuous in D if for every e > 0 there exists a 6 > 0 such that, for all f E F.
1(w) — f (z)I
1 and E = oc, F vanishes identically in T by 4.3.4b). In particular,
F(n + 1) =
t h(t) • tndt = 0
for all n E N.
Hence, by the corollary, t h(t) is identically zero in I. Historical remark. In his 1914 paper (Mu], C. H. Mfintz discovered the following generalization of the Weierstrass approximation theorem:
Let k,, be a real sequence such that 0 < k, < • • • k n < • and E 1/k, = oc. Then every continuous function on [0,1] can be approximated uniformly by functions of the form E7a,,x k u .
Miintz derived his identity theorem from this (arguing as in the proof of the corollary above). The function-theoretic proof given here is due to T. Carleman (rah especially p. 15). The converses of Mfintz's identity theorem and approximation theorems are true; cf. [Rud], pp. 312-315. — Elementary proofs of Miintz's approximation theorem can be found in the papers of L. C. G. Rogers [Ro] and M. v. Golitschek [G].
§5.
Consequences of a Theorem of Hurwitz
In this section we gather some properties of limit functions of sequences
of holomorphic functions that will be needed later. There is no connection with the theorems of Montel and Vitali; rather, the following lemma, obtained in 1.8.5.5, is our starting point. Lemma (Hurwitz). Let the sequence f„ E 0(G) converge compactly in G to a nonconstant function f E 0(G). Then for every point c E G there exist an index n e E N and a sequence c„ E G, n > nc , .such that lim c „ = c and
f „(c„) = f (c), n >
This lemma, which is a special case of a more general theorem of Hurwitz (cf. 1.8.5.5), has important consequences. The following is well known. Corollary. Let the sequence f„ E 0(G) converge compactly in G to f E 0(G). If all the functions f n are nonvanishing in G and f is not identically zero, then f is nonvanishing in G.
§5. Consequences of a Theorem of Hurwitz
163
Proof. We may assume that f is not constant. Then if f had a zero e G G, by Hurwitz almost all the f„ would have zeros cr, E G.
Another corollary follows from this one. Corollary. If the sequence fn E 0(G) converges compactly in G to a nonconstant function f E 0(G); then the following statements hold. (1) If all the images fn (G) are contained in a fixed set A C C, then f (G) C A. C. C are injective, then so is f : G (2) If all the maps f„. : G C are locally biholomorphic, then so (3) If all the maps f,, : G is f : G C. Proof. ad (1). Let b E C\ A. Since MG) C A, every function f,, - b is 0, the corollary implies that f - b is nonvanishing in G. Since f - b nonvanishing in G. This means that b f (G). It follows that f (G) C A. ad (2). Let c E G. By the injectivity of all the fn , the functions f,„ - f,1 (c) 0, the corollary implies that are all nonvanishing in G\{c}. Since f f (c) f - f(c) is nonvanishing on G\Icl. Hence f(z) f (c) for all z E G\{c} Since c E G was chosen arbitrarily, the injectivity of f follows. ad (3). The sequence fa.' of derivatives converges compactly in G to f'. Since f is not constant, J.' is not the zero function. By the local biholomorphy criterion 1.9.4.2, all the f are nonvanishing in G. By the corollary, f' C is also nonvanishing in G; hence — again by 1.9.4.2 — the map f : G is locally biholomorphic. 0 Remark. If A is a domain such that A = A (a disc, for example), then statement (1) follows directly from the open mapping theorem 1.8.5.1. (Prove this.) The following formulation of statements (1) and (2) is used in the proof of the Riemann mapping theorem: Hurwitz's injection theorem. Let G and G' be domains, and let f„ : G -■ G' be a sequence of holomorphic injections that converges compactly in G to a nonconst ant function f E 0(G). Then f(C) C G' and the induced map f : G -> G' is injective.
We also note a supplement to statement (2), which will be used in 9.1.1.
(2') If all the maps f„ are injective and lirn f,, (b,,) = f (b), where b,„ b E G, then lim b„ = b. In particular, lirn f I (a) = f -1 (a)
for every a E f (G)
n n f„(G). n>0
164
7. The Theorems of Montel and Vitali
Proof. If lirn bn were not equal to b, there would exist E > 0 and a subsequence bn, of the sequence b„ with bn, B := B(b). By the injectivity of all the fn , the sequence fn , — f(b) would then be nonvanishing in B. Its limit f — f (b) would therefore also be nonvanishing in B; but this is impossible. Thus bn = b.
Bibliography [Bli
BLASCHKE, W.: Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, Ber. Verh. Königl. Siichs. Ces. Wiss. Leipzig 67, 194-200 (1915); Ces, Werke 6, 187-193.
[Bu]
BURCKEL, R. B.: An Introduction to Classical Complex Analysis, vol. 1, Birkhauser, 1979.
[Cal
CARLEMAN, T.: Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. for Mat. Astron. Fys. 17, no. 9 (1923).
[CL)
CARATHEODORY, C. and E. LANDAU: Beitriige zur Konvergenz von Funktionenfolgen, Sitz. Ber. lanigl. Preuss. Akad. Wiss., Phys.-math. KI. 26, 587-613 (1911); CARATHEODORY's Ges. Math. Schriften 3, 13-44; LANDAU's Coll. Works 4, 349-375.
[G ]
GOLITSCHEK, M. v.: A short proof of Miintz's theorem, Journ. Approx. Theory 39, 394-395 (1983).
[HS]
HERMITE, C. and T.-J. STIELTJES: Correspondance d'Hermite et de Stieltjes, vol. 2, Gauthier-Villars, Paris, 1905.
[Hi]
HILBERT, D.: Über das Dirichletsche Prinzip, Jber. DMV 8, 184-188 (1899); Ces. Abh. 3, 10-14.
PI
JENTZSCH, R.: Untersuchungen zur Theorie der Folgen analytischer Funktionen, Acta Math. 41, 219-251 (1917).
[K]
KOEBE, P.: Ueber die Uniformisierung beliebiger analytischer Kurven, Dritte Mitteilung, Nachr. Kiinzgl. Ces. Wiss. GOttingen, Math. phys. Kl. 1908, 337-358.
[Li]
LINDEL6F, E.: Démonstration nouvelle d'un théorème fondamental sur les suites de fonctions monogènes, Bull. Soc. Math. France 41, 171-178 (1913).
[LR]
LAWNER, K. and T. RAD6: Bemerkung zu einem Blaschkeschen Konvergenzsatze, Jber. DMV 32, 198-200 (1923).
[Moi]
MONTEL, P.: Sur les suites infinies de fonctions, Ann. Scz. E. Norm. Sup. 24, 233-334 (1907).
[MO2]
MONTEL, P.: Leçons sur les familles norm'ales de fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927; reissued 1974 by Chelsea Publ. Co., New York.
[Miil
MÛNTZ, C. H.: Über den Approxirnationssatz von Weierstrai3, Math. Abh. H. A. Schwarz gewidmet, 303-312, Julius Springer, 1914.
§5. Consequences of a Theorem of Hurwitz
165
[0]
OSGOOD, W. F.: Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals, Ann. Math., 2nd ser., 3, 25-34 (1901-1902).
1P1
PORTER, M. B.: Concerning series of analytic functions, Ann. Math., 2nd ser., 6, 190-192 (1904-1905).
[Rol
ROGERS, L. C. G.: A simple proof of Miintz's theorem, Math. Proc. Cam-
bridge Phil. Soc. 90, 1 3 (1981). -
RUDIN, W.: Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. [Run]
RUNGE, C.: Zur Theorie der analytischen Functionen, Acta Math. 6, 245 -
248 (1885). STIELTJEs, T.-J: Recherches sur les fractions continues, Ann. Fac. Scz. Toulouse 8, 1-22 (1894).
ISZI
SAKS, S. and A. ZYGNIUND: Analytic Functions, 3rd ed., Elsevier, Warsaw,
1971. [V]
VITALI, G.: Sopra le serie de funzioni analitiche, Rend. 1st. Lombardo, 2. Ser. 36, 772-774 (1903) and Ann. Mat. Pur. Appl., 3. Ser. 10, 65-82 (1904).
8 The Riemann Mapping Theorem
Zwei gegebene einfach zusammenliiingende ebene Flkhen kiinnen stets so auf einander bezogen werden, daf3 jedem Punkt der einen Ein mit Ulm stetig fortriickender Punkt der andern entspricht und ihre entsprechenden kleinsten Theile ähnlich sind. (Two given simply connected planar surfaces can always be related to each other in such a way that every point of one corresponds to one point of the other, which varies continuously with it, and their corresponding smallest parts are similar.) — B. Riemann, 1851
Since Riemann, the problem of determining all domains in the plane that are biholomorphically (= conformally) equivalent to each other has been one of the main interests of geometric function theory. Existence and uniqueness theorems make it possible to study interesting and important holomorphic functions without knowing closed analytic expressions (such as integral formulas or power series) for them. Furthermore, analytic properties of the mapping functions can be obtained from geometric properties of the given domains. The Riemann mapping theorem — this chapter's epigraph — solves the problem of when simply connected domains can be mapped biholomorphically onto each other. In order to understand this theorem, we familiarize ourselves in Section 1 with the topological concept of a "simply connected domain." Intuitively, these are domains without holes; that is, domains in which every closed path can be continuously contracted to a point (i.e. is
168
8. The Riemann Mapping Theorem
null ho7notopic). We discuss two integral theorems; the result. that will be crucial is the following: Simply connected domains G in C are homologically simply connected: f d( = O for all f e 0(G) and all piecewzse continuously differentiable paths -y in G.
Readers primarily interested in the Riemann mapping theorem may skip Section 1 on a first reading and think of the concepts "simply connected" and "homologically simply connected" as equivalent. It took many years and the greatest efforts to prove Riemann's assertion. 1 as C. Neumann, H. A. Schwarz, H. Poincaré, D. Suchmateins Hilbert, P. Koebe, and C. Carathéodory worked on it. Finally, in 1922 ) the Hungarian mathematicians L. Fejér and F. Riesz gave their ingenious proof by means of an extremal principle. In Section 2 we reproduce Carathéodory's variant of the Fejér-Riesz proof. In Section 3 we give a detailed account of the history of the mapping theorem. Section 4 contains supplements to the mapping theorem, including a Schwarz lemma for simply connected domains.
§1.
Integral Theorems for Homotopic Paths
In star-shaped domains, integrals of holomorphic functions over paths with fixed initial and terminal points do not depend on the choice of path. This path independence remains valid for arbitrary domains as long as the path of integration is "only continuously deformed." Precisely what this means will be explained in this section. Two notions of homotopy, which will be introduced in Subsections 1 and 2, are fundamental. To each notion of homotopy corresPonds a version of the Cauchy integral theorem. The proofs of these integral theorems are elementary but rather technical; all they use from function theory is that holomorphic functions in discs have antiderivatives. In Subsection 3 we show that null-homotopic paths are always null homologous, but that the converse is not true in general. The basic concept of a "simply connected domain" is introduced and discussed in detail in Subsection 4. 1. Fixed-endpoint homotopic paths- Two paths -y, ;y- with the same initial point a and terminal point b in a metric (more generally. topological) space X are called fixed-endpoint homotopic in X if there exists a I Incidentally, L. Ahlfors writes: "Riemann's writings are full of almost cryptic messages to the future. For instance, Riemann's mapping theorem is ultimately formulated in terms which would defy any attempt of proof, even with modern methods."
§1. Integral Theorems for Hmnotopic Paths
continuous map (*) 0(0, t) = -y(t)
:I x
X, (s, t)
169
(s, t). such that, for all s, t E I,
and V(1, t) = -7 (t) ,
(s , 0) = a and
V) ( s , 1) = b.
The map 1/) is called a homotopy between -y and '5%. For every s E I, : I —> X, t 1—> lb (. 5 , t) , is a.path in X from a to b; the family (-y Ei is a "deformation" of the path -7 = -yo into the path -y 1 = We note (omitting the simple proof) that "being fixed-endpoint homotopic in X" is an equivalence relation on the set of all paths in X from a to b.
FIGURE 8.1. The importance of lie concept of homotopy just introduced is shown by the
Cauchy integral theorem (first homotopy version). Let -y,
be pzecewise continuously differentiable fixed-endpoint homo topic paths in the domain G c C. Then
fy
f d( = f f d(
for all f E 0(G).
Observe that the paths -ys , 0 < s < 1, need not be piecewise continuously differentiable. The idea of the proof can be explained quickly. The rectangle I x I is subdivided into rectangles I,, in such a way that the images 1//(/ ),) lie in discs C G (Figure 8.1). By the integral theorem for discs, the integrals of f along all the boundaries 07,15(4,) are zero. Hence the integrals of f are equal along sufficiently close paths -y,, -y, , (see figure 8.1). -- The technically rather tedious details are given in Subsections 5 and 6; they can be skipped on a first reading.
2. Freely homotopic closed paths. Two closed paths 'y, in X are said to be freely homotopic in X if there exists a continuous map x I —4 X
170
8. The Riemann Mapping Theorem
with the following properties:
(*)
0(0,0 = -y(t) and '(s, O) = 14, (s, 1)
tp(1,t) = 7y(t)
for all t G for all s E I.
0(s. t) are closed; their initial points Then all the paths -y, : I X, t trace the path 6 : I —) X, t 1—) 1,1)(t, 0) in X (Figure 8.2). The paths 1, and 6+ — 6 have the same initial and terminal points. The following statement is intuitively clear.
y(0)
FIGURE 8.2.
(1) If -y and 5 are freely homotopic in X, then the paths -y and 6 +5 —6 are fixed-endpoint homotopic in X. Proof. For every s E I, set x, := 61[0, s] + -y, — 61[0, s]. A parametrization can be chosen (!) such that x : I x I X, (s, t) x,(t), is continuous 0 and hence a fixed-endpoint homotopy between -y and 6 + — 6. Cauchy integral theorem (second hoinotopy version). Let -y, be piecewise continuously differentiable closed paths in the domain G c C that are freely homotopic in G. Then f
d(
= f f d(
for all J E 0(G).
The proof follows trivially from (1) and Theorem 1 if, in addition, 6 is piecewise continuously differentiable; for then fd(= f
fd(= f
fd(— f fd(=
6+ 7i-6
The proof of the general case is completely analogous to that of Theorem 1 (cf. Subsections 5* and 6*, where V) is now it "free homotopy" between and 3. Null homotopy and null homology. A closed path -y in G is said to be null homotopic in G if it is freely hornotopic to a constant path (point
path). By 2(1), this holds if and only if -y is fixed-endpoint homotopic in
§1. Integral Theorems for Homotopic Paths
171
G to the constant path t i. -y(0). The next statement follows immediately from Theorem 2. Proposition. Every piecewise continuously differentiable closed path 7 that is null hornotopie in G is null homologous in G:
fy fd = 0
for all f E 0(G).
The interior Int -y of every such path also lies in G; cf. 1.9.5.2. Null homology is a consequence of null homotopy, but the converse is false. In G := CV-1, 1}, for instance, consider the boundaries 'y2 , y3, 1'4 of the discs B1 (-1), B 1 (1), B2(-2), B2(2), each with 0 as initial and terminal point (Figure 8.3). Then one can prove:
FIGURE 8.3.
The closed path -y := -yi — -y3 — -y2 + -y4 is null homologous but not null homotopic in G. It is easy to see that -y is null homologous in G:
lut -y = [B2 (-2)\B 1 (-1)] U [B2(2) \B i (1)] c G
(Figure 8.3).
It is also immediately clear from the figure that 1, is not, null homotopic in G: every deformation from to a point must certainly pass through the omitted points 1 or —1. To give a clean argument is harder; one can argue, for example, as follows. Choose a function f holomorphic at 0 which can be continued holomorphically along every path in G but whose continuation along -y does not lead back to f (one such function is f (z) = N /log 1(1 + z), where f(0) iVlog 2). Then, by the monodromy theorem, -y is not null homologous in G. The argument shows that the fundamental group of the twice-punctured plane is not abelian. 4. Simply connected domains. A path-connected space X is called simply connected if every closed path in X is null homotopic. Clearly, this occurs if and only if two arbitrary paths in X with the same initial and terminal points are always fixed-endpoint homotopic in X.
172
8. The Riemann Mapping Theorem
(1) Every star-shaped domain G in C (or IV ) is simply connected. Proof. Let c e G be a center of G. If -y is a closed path in G, the continuous
map G,
(s, t)
0(s, t) := (1 — s)-y(t) + se
is a free homotopy between -y and the constant path c. In particular, all convex domains, of which the plane C and the unit disc E are special cases, are simply connected. — Simple connectivity is a topological invariant.: (2) If X X' is a homeomorphism, then X' is simply connected if and only if X is. In particular, every domain in C that can be mapped topologically onto E is simply connected. The (Cauchy) function theory of a domain G is simplest when G is homologically simply connected; that is, when every closed path in G is null homologous in G. The next result follows immediately from Proposition 3. Proposition. Every simply connected domain G in C is homologically simply connected.
This proposition will be used to prove the Riemann mapping theorem. In the course of the proof, it will be shown that the converse of the proposition also holds; cf. 2.6. Later, by means of Runge theory. we will see that simply connected domains G in C are also characterized by having no holes (= compact components of C\G); cf. 13.2.4. Historical note. Riemann introduced the concept of simple connectivity in 1851 and recognized its great significance for many function-theoretic problems; he defines it as follows ([Rie], p. 9): "Erne zusammenheingende Fldche heifit, wenn sic durch jeden Querschnitt in Stiicke zerfallt, eine ein-
fach zusammenhangende." (A connected surface is called simply connected if it is divided into pieces by every cross cut.) By cross cuts he means, in this context, "Linien, welche von einem Begrenzungspunkte das innere einfach keznen Punkt mehrfach — bis zu einem Begrenzungspunkte durchschneiden" (lines which cut across the interior simply — no point more than once — from a boundary point to a boundary point). It is intuitively clear that this definition does in fact describe simply connected domains. 5*. Reduction of the integral theorem 1 to a lemma. We first prove:
:IxI G is continuous, there exist numbers s o , s i , . . . , s m , t o , t , . , t , with 0 = so < s 1 < • • • < .s„, = 1 antiO = to < ti < • • < t = 1, such that for every rectangle I,,, := [sp , sp+1]x[t,,,t,,+1 ] the ip-intage 0(100 is contained in an open disk B p, c G, 0 < < m, 0 < u < n. (1) If
Proof. We cover G by open discs 133 , j E J. By the continuity of 71), the family {7,b - 1 (B3 ), j e J} is an open cover of/x/(where/x/cWis
1. Integral. Theorems for Homotopic Paths
173
equipped with the relative topology). There exists a cover of ./ x I by open rectangles R parallel td- the axes such that each R lies in a set tirl (.8 ? )• Since I x I is compact, it is already covered by finitay many such rectangles Rk. Each rectangle Tir, is of the form (a, x [T, 71, with o < a < a' < 1, 0 < < q-' < 1. Arranging and enumerating all the , sm , t0 ,... ,t„, with a and 7- appearing here, we obtain numbers so , = I and 0 = to < ti 0. 6. Equivalence theorem. The following statements about a domain G in C are equivalent:
i) G is homologically simply connected. ii) Every function h,olomorphic in G is integrable in G. iii) For all f E 0(G) and every closed path 7 in G, \
ind (z)fz/ -y
1 f 27ri
tg. –
z
zE
1'Y!
iv) The interior Int -y of every closed path 7 in G lies in G. y) Every unit in 0(G) has a holomorphic logarithm in G. vi) Every unit in 0(G) has a holomorphic square root in G. vii) Either G = C or G can be mapped biholonzorphically onto E. viii) G can be mapped topologically onto E. ix) G is simply connected. Proof. Equivalences 1) through vi) are known from 1.9.5.4. — vi) vii). This is Theorem 1 (where it is now unnecessary that 0 E G). vii) viii) ix). Trivial (note the introduction to this section and 1.4(1) and (2)). — ix) i). This is Proposition 1.4.
This theorem is an aesthetic peak of function theory. The following turn out to he equivalent: — topological statements (simple connectivity); — analytic statements (Cauchy integral formula); algebraic statements (existence of a square root).
Each of these statements implies that one is actually looking at C or E. The list of nine equivalences can be considerably and nontrivially lengthened. Thus one can add: x) G has no holes. xi) Every function in 0(G) can be approximated compactly in G by polynomials. (G is a Runge domain.) xii) G is homogeneous relative to Aut G and G Cx . xiii) There exists a point a E G with infinite isotropy group Aut.G. xiv) The monodromy theorem holds for G. That ix), x), and xi) are equivalent is proved in 13.2.4. The equivalence xii) ix) is proved in 9.1.3 for bounded domains "with smooth boundary components." The last two equivalences will not be pursued further.
§3. On the History of the Riemann Mapping Theorem
§3.
181
On the History of the Riemann Mapping
Theorem The names of many mathematicians are inseparably linked to the history of the Riemann mapping theorem: Carathéodory, Courant, Fejér, Hilbert, Koebe, Riemann, Riesz, Schwarz, Weierstrass.
There are three different ways to prove the theorem, using — the Dirichlet principle,
—methods of potential theory, or —the Fejér-Riesz extrema! principle.
We describe the principal steps in the development of each approach. 1. Riemann's dissertation. Riemann stated the mapping theorem in his dissertation in 1851, and sketched a proof tailored to bounded domains with piecewise smooth boundaries ([Rie], p. 40). He uses a method of proof that ties the problem of the existence of a biholomorphic map G E to the Dirichlet boundary value problem for harmonic functions. Riemann solves the boundary value problem with the aid of the Dirichlet principle, which characterizes the desired function as that function y) with given boundary values for which the Dirichlet integral
f jG ( + 4)(1:My has the smallest possible value. Riernarm's revolutionary ideas are not accepted by his contemporaries; the time is not yet ripe. An awareness of the mapping theorem as an existence theorem is missing. Only after his death does Riemann find public relations agents in H. A. Schwarz, L. Fuchs, and, above all, F. Klein; cf. F. Klein: "Riemann und seine Bedeutung fiir die Entwicklung der modernen Mathematik," a lecture given in 1894 (Ges. Math. Abh. 3, pp. 482-497). What did the philosophy faculty of the venerable Georgia Augusta at Gottingen think of Riemann's dissertation? E. Schering reports on this in a eulogy, "Zinn Gedàchniss an B. Riemann," delivered before the Gottingen Academy on 1 December 1866 but not published until 1909, in Schering's Ces. Math. Werke 2, p. 375, Verlag Mayer und Müller Berlin. This eulogy remained almost unknown until it appeared recently in the new edition of Riemann's Werke edited by R. Narasimhan, pp. 828-847. Springer und Teubner 1990; cf. p. 836 in particular. In Figure 8.5, we reproduce a copy of parts of the Riemann-Akte Nr. 135, with the kind permission of the Göttingen University archives. The dean of the faculty,
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§3. On the History of the Riemann Mapping Theorem
183
Ewald (1803-1875, an evangelical theologian, orientalist, and politician, one of the Göttingen Seven), asks Gauss for an expert opinion; the dean finds "das Latein in dem Gesuche und der Vita [von Riemann] ungelenk und kaum ertriiglich" (the Latin in the application and vita [of Riemann] clumsy and almost unbearable). Gauss, in his terse commentary. does not say a word about the content of the work. The reviewer, known to be sparing of praise, does speak of "griindliche und tief eindringende Studien in demjenigen Gebiete, welchem der darin behandelte Gegenstand angeh6rt" (profound and deeply penetrating studies in that area to which the subject treated there belongs), of "strebsame ächt mathematische Forschungsgeiste" (industrious and ambitious genuine mathematical spirit of research), and of "riihmliche productive Selbstth&igkeit" (commendable productive creativity). He thinks that "der grater Theil der Leser m6chte lArolil in einigen Theilen noch eine griiBere Durchsichtigkeit der Anordnung wanschen" (most, readers may well wish in some places for greater clarity in the presentation); he summarizes, however, by saying: "Darns Ganze ist eine gediegene werthvolle Arbeit, das Maati der Anforderungen, welche man gew6hnlich an Probeschriften zur Erlangung der Doctorwiirder stellt, nicht bla erfiillend, sondern wen iiberragend." (The whole is a solid work of high quality, not merely fulfilling the requirements usually set for a doctoral thesis. but far surpassing them.) Gauss did not suggest a grade; a third of his letter concerns a time and date that would be convenient for him, not too early in the afternoon, for the oral exam.
2. Early history. In his 1870 paper [Wei], Weierstrass temporarily pulled the rug out from under Riemann's proof with his criticism of the Dirichlet principle, by showing through examples that the existence of a minimal function is by no means certain. Around the turn of the century Hilbert weakened this criticism by a rigorous proof of the Dirichlet principle to the extent required by Riemann ([Hi]. pp. 10-14 and 15 37). Since then it has regained its place among the powerful tools of classical analysis; cf. also [Hi], pp. 73-80, and the 1910 dissertation of Courant [Cou]. In the meantime. other methods were developed. C. Neumann and H. A. Schwarz devised the so-called alternating method; see the encyclopedia articles [Lich] and [B] of L. Lichtenstein and L. Bieberbach, especially [Lich], §48. The alternating method allows the potential-theoretic boundary value problem to be solved for domains that are the union of domains for which the boundary value problem is already known to be solvable. With this method. which also uses the Poisson integral and the Schwarz reflection principle, Schwarz arrived at results which finally culminated in the following theorem ([Schwl, vol. 2, passim): If G is a simply connected domain bounded by finitely many real analytic curves which intersect zn angles not equal to 0, then there exists a topological map from G onto R. that maps G biholomorphically onto E. In.Schwarz's time. statements of this kind were considered the hardest. in all analysis. — Simply connected domains with arbitrary boundary were
184
8. The Riemann Mapping Theorem
first treated in 1900 by W. F. Osgood [Osg]; Osgood's research is based on earlier developments by Schwarz and Poincaré. 3. From Carathéodory-Koebe to Fejér-Riesz. The proof in 2.2-4 is an amalgam of ideas of C. Carathéodory, P. Koebe, L. Fejér, and F. Riesz. All the methods of proof known up to 1912 use the detour via the solution of the (real) boundary value problem for the potential equation Au = 0. In 1912 Carat héodory had the felicitous idea, for a given domain G, of mapping the unit disc E onto a sequence of Riemann surfaces whose "kernel" converges to G; the sequence fn itself then converges compactly to a biholornorphic map f : E -4 G (cf. [Ca l ], pp. 400-405, and Satz VI, p. 390). Thus for the first time, with relatively simple, purely function-theoretic means. "die Abbildungsfunktion (lurch em n rekurrentes Verfahren gewonnen [wurde], das bei jedem Schritt nur die Aufl8sung von Gleichungen ersten und zweiten Grades verlangt" (the mapping function [was] obtained by an iteration process, which at each step required only the solution of first- and seconddegree equations) (loc. cit., p. 365). Koebe could immediately, to a large extent, eliminate Carathéodory's auxiliary Riemann surfaces [Koe 3 , 4 ]: thus there emerged a very transparent constructive proof for the fundamental theorem, of conformal mapping. The construction of expansions described in Lemma 2.3 plays the central role ([Koe 4 ], pp. 184-185). We will present this beautiful Carathéodory-Koebe theory in detail in the appendix to this chapter and, in doing so, will also go more deeply into the "competition" between these two mathematicians. In 1922 L. Fejér and F. Riesz realized that the desired Riemann mapping function could be obtained as the solution of an extremal problem for derivatives. They had their stunningly short proof published by T. Rad6 in the recently founded Hungarian journal Acta Szeged. R.ad6 needed one full page to present it ([Ra], pp. 241-242); the Carathéodory-Koebe square root transformation was godfather to this "existence proof of the first water." Fej6r and Riesz consider bounded Q domains G and show: -
There exist p > 0 and a bzholomorphic map h : G and h'(0) = 1,
B(0) such that h(0) = 0
The key to the proof is furnished by the (nonempty) family
:= { f E 0(G) : f is bounded and injective, f (0) = 0, f' (0) = 11. Corresponding to p f E 711 < oc. there exists a sequence L E 71 that is bounded in G and satisfies lim IL Ic = p. By Montel, a subsequence converges compactly in G to some h E 0(G). We have h(0) = 0, W(0) = 1, and ING = p. By Hurwitz, h : G — ■ B p (0) is injective; in particular. it follows that h E N and p > 0. The surjectivity of h is now the crucial point: assuming that h(G) B(0). Fejér and Riesz use the Carat héodory -Riesz square root method to construct a function -it E 11 with 'hi,: < p: this contradicts the minimality of p.
4. Carathéodory's final proof. For the proof that Te(0) = 1, Fejér and Riesz must compute explicit derivatives: cf. [Ra ], pp. 241-242. In 1929 A.
§3. On the History of the Riemann Mapping Theorem
185
Ostrowski published a variant of the Fejér-Riesz proof in which "siimtliche Reehnungen - auch die Berechnungen der Nullpunktsableitung der Abbildungsfunktion — vermieden werden" (all computations — even the calculation of the derivative at zero of the mapping function — are avoided) ([0st], pp. 17-19). Ostrowski works with the fatnily ,F of 2.4 and begins by observing (substitute this for Proposition 2.4): If G
C is a Q -domain, then h(G) = E for every function h
(*).
.F with
1h' (WI = supfit(0)1 : f E
with f(G) E there exists by Lemma 2.3 an expansion g(G) E. Since ire(0)1 > 1 (see 1.1 of the appendix to this chapter), it follows that > Ig'(0)1 for g- :=KogE.F. The existence of a function h E F satisfying (*) now follows again from the fact that. by Montel, there exists a sequence h3 E .F with limit/ 31 (0)j = p := sup{ I f(0)1 : f E 7- } that converges compactly to some h E 0(G). Then Ih'(0)1 = p.. Since ,F 101 implies that A > 0, it follows from Hurwitz that h E F. Indeed, for every g :
-
When Ostrowski published his proof, he was unaware that not long before, in 1929, in the almost unavailable Bulletin of the Calcutta Mathematical Society, Carathéodory had presented a variant completely free of derivatives: "... durch eine geringe Modifikation in der Wahl des Variationsproblems [kann man] den Fe3ér-Rieszschen Beweis web wesentlich vereinfachen" (... through a minor modification in the choice of the variational problem, [one can] further simplify the FeJer-Riesz proof consi (1 ([Car], PP. 300-301). We have presented Carathéodory's version-erably) in 2.2-4. It is the most elegant proof of the Riemann mapping theorem. In the prevailing literature, however, up to the present, Ostrowski's version has won out over that of Carathéodory; the 1985 textbook [NI of R. Narasirnhan is an exception. In 1928 Carathéodory wrote the following on the history of the proof of the mapping theorem ([Ca l ], p. 300): "Nachdem die Unzuliinglichkeit des urspriinglichen Riemannschen Beweises erkannt worden war, bildeten für vielen Jahrzehnte die wundersch6nen, aber sehr urnstndlichen Beweismethoden, die H.A. Schwarz entwickelt hatte, den einzigen Zugang zu diesem Satz. Seit etwa zwanzig Jahren sind dann in schneller Folge eine Graf. Reihe von neuen kiirzeren und besseren Beweisen [von ihm selbst und von Koebe] vorgeschlagen worden; es war aber den ungarischen Mathematikern L. Fejér und F. Riesz vorbehalten, auf den Grundgedanken von Riemann zuriickzukehren und die L6sung des Problems der konformen Abbildung wieder mit der L6sung eines Variationsproblems zu verbinden. Sie wiihlten aber nicht em n Variationsproblem, das, wie das Dirichletsche Prinzip, auBerordentlich schwer zu behandeln ist, sondem em n solches, von dem die Existenz einer L6sung feststeht. Auf diese Weise entstand em n Beweis, der nur wenige Zeilen tang ist, und der auch sofort in alien neueren
186
8. The Riemann Mapping Theorem
Lehrbiichern aufgenommen worden ist." (After the inadequacy of Riemann's original proof was recognized, the exquisite but very intricate methods of proof developed by H. A. Schwarz were for many decades the only approach to this theorem. Then, since about twenty years ago, a great number of new shorter and better proofs [his own and Koebe's] have been proposed in rapid succession; but it remained for the Hungarian mathematicians L. F(jér and F. Riesz to return to Riemann's basic ideas and once again tie the solution of the conformal mapping problem to the solution of a variational problem. They did not, however, choose a variational problem that, like the Dirichlet principle, is extremely hard to handle, but rather one that is certain to have a solution. In this way a proof emerged that is only a few lines long and that was also immediately adopted by all new textbooks.) In fact, the Fejer-Riesz proof appears as early as 1927 in Bieberbach's Lehrbuch der Funktionentheorie, vol. 2, p. 5. The old proofs were forgotten: The better ls the enemy of the good.
5. Historical remarks on uniqueness and boundary behavior. In his dissertation, Riemann not only asserted the existence of a conformal map f : G 1 G2 between simply connected domains Gi , G2, but stated that f can moreover be chosen to be a topological map G G2: that, in particular, f maps the boundaries 0G1, 0G2 topologically onto each other (for Riemann, all boundaries are piecewise smooth). Riemann also had precise ideas about when f is uniquely determined ([Rie]. p. 40): Zn Einem innern Punkte und zu Einem Beyrenzpunkte Mann] der entsprechende beliebig gegeben werden; dadurch aber ist fur Punkte die Beziehung bestimmt. (The points corresponding aile to an interior point and to a boundary point [can] be assigned arbitrarily; doing this, however, determines the relationship for all points.) In 1884 Poincaré proved a uniqueness theorem that assumes nothing about the existence of the map on the boundary of G; cf. Lemme fondamental ([Po], p. 327). Poincaré's lemma is in modern language - nothing but the uniqueness theorem 2.5; our proof is essentially the same as Poinca.ré's (loc. cit., pp. 327-328). — The uniqueness problem played an important role in the history of conformal mapping theory. In 1912 Carathéodory, by determining that the uniqueness theorem is ultimately based on Schwarz's lemma, put the elegant finishing touches on these matters. Schwarz, in 1869, sharply separated the -prpblem of mapping a domain conformally onto a disc from the problem of extending this map continuously to the boundary. Carathéodory studied the extension problem from 1913 on and developed his penetrating theory of prime ends; cf. the first three papers in the fourth volume of his Gesammelte Mathematische
Schriften.
§3. On the History of the Riemann Mapping Theorem
187
ln order to formulate the high points of extension theory. we consider a biholomorphic map f : E G onto a bounded domain G. The following lemma is the starting point.
Extension lemma. The map f can be extended to a continuous map from R. to G if and only if the bowndary of G is a closed path (in other words, if C with y,-(01E) 0G). there exists a continuous map ;o : OE This lemma is used to prove
Carathéodory's theorem. The map f : E l; can be extended to a topological map from 17, onto G if and only tf the boundary of G is a closed Jordan curve (in other words, if there exists a topological map : 0E with .)(0E) = 0G).
C
A simple corollary is Schoenfiies's theorem on Jordan curves. which has nothing to do with function theory:
Every topological map from one Jordan curve onto another can be extended to a topological map from C onto itself. Details of this theory can be found in the book [Porn].
6. Glimpses of several variables. There is no obvious generalization of the Riemann mapping theorem to simply connected domains in C", > 1, even in the case n = 2. The polydise ((w. z) e c2 : < I. '1 21 < 1} and the ball {(w, z) C2 1w1 2 lz!2 < 1} are natural analogues of the unit disc; both domains are topologically cells, thus certainly simply connected. But Poincaré proved in [907:
There is no bihulomorphic map from. the ball onto the polydise. Simple proofs are given in [Ka], p. 8, and [Rani, p. 24. Then. even exist families of bounded domains of holomorphy G t , I E R, with boundaries ()G1 that are real analytic everywhere. such that all the domains Gi are diffeomorphic to the 4-dimensional cell but two domains Gf G arc biholomorphically equivalent only if t Positive statements eau be obtained if the automorphisms of the domain are brought into play. Thus, for example, É. Cartan proved in 19:l5
Every bounded homogeneous domain in C 2 can be mapped biliolom orphicalk onto either the ball or the bidisc. For n > 3 the situation is more complicated, even neous domains.
for hounded homoge-
188
§4.
8. The Riemann Mapping Theorem
Isotropy Groups of Simply Connected Domains
The automorphism group Aut G of all the biholomorphic maps of a domain G onto itself contains important information about the function theory of G. Two domains G, G' can be mapped biholomorphically onto each other only if their groups Aut G, Aut G' are isomorphic. In addition to automorphisms, we study inner maps of G; these are holomorphic maps from G to itself. The set Hol G of all the inner maps of G, with composition as group operation, is a semzgroup with Aut G as a subgroup. For every point a E G. the set Holo G of all the inner maps of G with fixed point a is a subsemigroup of Hol G. The set Auto G of the automorphisms of G with fixed point a is a subgroup of Hole G; Aut o G is also called the isotropy group of G at a. The map : Holo G
C, f 1
-
■
f '(a)
is fundamental for the study of Holo G and Auto G. It is multiplicative (by the chain rule): (f o
(a) = (a)g' (a), f, g E Holo G;
in particular, a induces a homomorphism Aut„ G Cx of the group Aut„ G into the multiplicative group Cx In Subsection 1, we describe a for four special domains. In Subsection 2, a is studied for simply connected domains C. The tools for doing this are the Riemann mapping theorem and the Schwarz lemma, which we use in the following form: MI 0, is chosen
190
8. The Riemann Mapping Theorem
as in 2.5. The number p = p(G, a) is called the mapping radius of G relative to a. Thus
p(G,a) =
(1)
h := mapping function of 2.5.
(a),
We also set p(C, a) := oc for all a E C. Monotonicity theorem. If
6,
G are simply connected and G C G, then
p(d, a) < p(G,a)
(2)
If there exists a point b G
for all
a
c G.
G such that p(d,b) = p(G,b), then
a = G.
E, : G ZE be chosen as in 2.5. Then Proof. Let G C and let h : G g := h' h E Holo G. Since g' (a) = (a)Iti' (a) > 0, 2(1) implies that g'(a) < 1. Hence h'(a) < -11/(a), and (2) follows from (1). It follows from p(a, b) = p(G,b) that h' (b) = (b); hence g' (b) = 1 for the maps corresponding to b. Now 2(1) and Theorem 2 imply that o h = idG; hence h 1 (E) = G, i.e. = G.
a
Exercises
1) Compute p(G , a), a E G, in the following cases:
a) G := Br(c), E C, r > 0; b) G := = upper half-plane; c) G := {z = re
EC:r >0 arbitrary, 0 < (12 < (po, where (po E
(0,21r1}. 2) If G is simply connected and g G —+ C is a holornorphic injection, then p(g(G), g(a)) =
3) Let G
(a)lp(G , a)
for all
a E G.
C be simply connected and let a E G. Prove:
a) 'fl u p(G , a) for every f E 0(G) with f(a) = 0, f (a) = 1. b) Equality holds in a) if and only if f maps G biholomorphically onto B p (0) ("minimum-maximorum principle"). Bibliography for Chapter 8: See p. 201
§1. Simple Properties of Expansions
191
Appendix to Chapter 8: Carathéodory-Koebe Theory The Fejér-Riesz-Carathéodory proof is not constructive: No rule is given (in 8.2.4) telling how the sequence f, with lima f„(p) 1 = i is to be constructed, and absolutely nothing is said about how to find the subsequence hi of the sequence fa. A proof free of these defects was given in 1914 by P. Koebe, who applied ideas of Carathéodory: By expansion, the domain is mapped successively onto subdomains on E in such a way that these subdomains exhaust the unit disc. Koebe obtains the expansion maps by elementary means, solving a quadratic equation and determining a boundary point with minimal distance from the origin; his expansion sequences converge — though slowly to the desired biholomorphic map G E; no passage to subsequences is necessary. In Section 1 we discuss the expansions used by Koebe; in Section 2 we describe the Carathéodory-Koebe algorithm and apply it to the special expansion family /C2 In Section 3 we construct other families suitable for the algorithm. •
P.
Simple Properties of Expansions
We begin with a simple expansion lemma (Subsection 1), which is fundamental for the considerations of this appendix. In Subsection 2 "admissible" expansions are discussed. In Subsection 3, as an example of such expansions, we study the "crescent expansion."
1. Expansion lemma. If G is a domain in C with 0 E G, then r(G) :=
E R: Bt (0) c GI = d(0,aG)
is called the inner radius of G (relative to the origin). We have 0 < r(G) < po; when r(G) cc, there always exist boundary points a E OG with i al = r(G). The following monotonicity property of inner radii with respect to holomorphic maps is crucial for our purposes in this appendix. Expansion lemma. Let f, g E 0(G) be nonconstant and let the map g: G C be injective. Assume moroever that f(0) = 0 and if(z)i > for all z E G. Then r(f (G)) > r(g(G)). Proof. Let B = B,(0) C g(G), s < oc. It suffices to prove that B C f(g -1 (B)). This will follow if we prove that ibi > 8 for every point b E af( 9,-1(B)) . We set h := fog -1 . There exists a sequence a„ E B with a := lint a„ E and b = lim h(a„) . Then loi > s, since otherwise a E B; thus b = h (a) E Ig(g -1 (w))1 = 1w1 holds h(B). Now the inequality ih(w)i = I f(g -1 (w))1 for all w E B. Hence bi= lim > lim Iai = l ai s.
192
8. The Riemann Mapping Theorem
Now let G c E and 0 E G. Then r(G) < 1. According to 8.2.3, holomorphic injections t : G E with K(0) = 0 and its(z)1 > z. z E G\{0}, are called expansions of G. It is trivial that
(1) -roti:G—E is an expansion if k : G
E and R. :
Eare
expansions with G D K(G).
The following is important: Proposition. If 1.; : G
E is an expansion, then1K! (0)1> 1 and r(K.(G))>
T(G)
Proof. We have K(z) = zf (z), with f E 0(G). Since If(z)1 > 1 in GM0 } , we also have f(0)1 > 1 (minimum principle): hence IK/ ( 0 )1 = If( 0)1 > 1. -- The inequality r(k(G)) > r(G) follows immediately from the expansion lemma. Warning. The reader should not think that K(G) D G for all expansions. The crescent expansions are instructive counterexamples: see Subsection 3. One often speaks of expansions if, besides ri(0) = O. only IN(z)I > Izi holds. In that case, the proposition is true with IK'(0)1 > 1.
2. Admissible expansions. The square root method. A Q-domain G E). is called a Koebe domain if G c E (this means that G C E but G Then 0 < r(G) < 1. An expansion K of a Koebe domain is called admissible if K (G) is again a Koebe domain. The next statement follows trivially from 8.2.1(1). (1) Every expansion is admissible.
K :
G
E of a Koebe domain such that K(G)
E
Admissible expansions have already appeared in Lemma 8.2.3. We state matters more precisely here: Theorem (square root method). Let G be a Koebe domain and let c E E. c2 V G. Let v E 0(G) be the square root of ge2IG with v(0) = c. Let d : E E be a rotation about O. Then s o g. o v is an admissible expansion of G and
(2)
1 Its-1 (0)1
+ 1(. 12 ,
21c1
•
Proof. By Lemma 8.2.3, ge o u : G E s an expansion; hence so is K: G E. If we had k(G) = E, it would follow that v(G) = E and (because q(.2 = v2 ) that gc 2 (G) = E, i.e. G = E. But this is impossible. By (1), K is therefore admissible. Equation (2) is equation 8.2.3(4).
§1. Simple Properties of Expansions
193
In particular, for every Koebe domain G the square root method yields E with W(0) > 1; these "normalized" an admissible expansion tç : G theory; cf. 2.3. CI in Carathéodory-Koebe leading role expansions play a The following representation of the square root y is useful in computations:
(3) Set G* := ge2(G). Then = q o gc2, where q E 0(G*) with q2 = zIG* , q(c 2 ) = c. For all expansions k constructed in the theorem above,
(4)
r(G) < r(K(G))
(sharpened version of Proposition I)
E\K(G) always has interior points in E.
(5)
Proof. ad (4). Let bEEn a(K(G)) with Ibi = r(K(G)). There exists a sequence r(G). By Zn E G\101 with lim k(z n ) -= b and a := lim zn E 0G. Then lai Lemma 8.2.3, id = o tç, where ipc E 0(E) and lc(z)1 < Izi for Z E E\{01. It follows that zn = .0,-(k(z.)); hence a =1,b,(b), so lai < Ibi and r(G) < r(K(G))• ad (5). Since 19, gc , go E Ant E, it suffices by (3) to show that E\q(G*) has interior points in E. But this is clear: (-q)(G*) C E and q(G) n (- q)(G*) = 0 (the second statement because 0 fl G* — compare the proof of 8.2.2). • 3* The crescent expansion.
All slit domains G, := EMt2 , 1) , 0 < t < 1, are Koebe domains. The effect of the admissible expansion K := gt o v on Gt is surprising: (1) The image domain k(G t ) is the "crescent" E\K , where K is the closed disc about p (1 + t2 )/2t with t E aK (Figure 8.6). In particular, K(G t ) 75 G.
Proof. By 2(3),
k
:
G -4 - k(G) can be factored as follows: g,
g,2
lit
FIGURE 8.6.
Here G* = gt2 (G t ) = 0] and H := q(G* ) {z E E : Re z > 0}. Since g(t) = 0, it thus suffices to show that gt (01-1) is the boundary of JEW. Since gt (01E) = 0E, the functions gt map the semicircle in aH onto
194
8. The Riemann Mapping Theorem
the arc in 5E that goes from ( := gt (i) through -1 = gt (1) to --(.7 := gt (-0 (note that Im < 0). The image under gt of the line iR is the circle L through t = gt (0) (punctured at 1/t) that intersects aE orthogonally at and -(" (conformality). The equation for L is thus lz - m 2 = 7n2 - 1, with TIL > 1 (by Pythagoras; the tangents to OE at (, ç are perpendicular to the radii and intersect each other on R). Since t E L. it follows that m = p: hence L = (010\lt -1 1. This proves that gt (01-1) = a(E\K). Remark. The crescent expansion is discussed from a computational point of view in [PS], Problem 90, Part IV.
§2.
The Carathéodory-Koebe Algorithm
If a nonempty set D(G) of admissible expansions is assigned (according to some rule) to every Koebe domain G we call the union D := UD(G), where G runs through all Koebe domains, an expansion family. With the aid of such families, "expansion sequences" can be assigned to every Koebe domain G in various ways. We set Go := G and choose n o E D(Go). Since rço is admissible, G I := n o (G0 ) is a Koebe domain; we can therefore choose n i E D(G). Since G2 := ki (GI) is again a Koebe domain, we can continue and (recursively) determine Koebe domains G„ and expansions Kr, E D(G7,) with K(G) = G„ ±1 , n E N. Then, by 1(1), h„ :=
o
Km _i 0 • • 0
ic0
: G -4 E,
h, = K T, o h„_ 1
is an admissible expansion. The procedure just described is called the Carathéodory-Koebe algorithm and the sequence h„ thus constructed an expansion sequence (for G relative to D). It will be shown that "correctly chosen" expansion sequences converge to biholomorphic maps h: G E.
1. Properties of expansion sequences. For all expansion sequences h, : G E: h'n (0) =
k(0),
0 Ihn4-1(z)1 >
(z)1 > • • • > I ho(z)I > Izi, z E
r(lin (G)) < r(h,+I (G)), li1T
G\fol,
limr(h„(G)) < 1,
I ten (o) 1 = 1.
Proof. Statements (1) and (2) follow immediately from the equation h„ = h„._ 1 , where Proposition 1.1 is used in the proof of (2). To prove (3).
K0
§2. The Carathéoclory-Koebe Algorithm
195
let Bt (0) C G, t > O. Then every map E —+ E, z h„(tz), is holomorphic. Since h7 (0) = 0, it follows from Schwarz that Ih(0)1 < lit. The sequence thin (0)1 is thus bounded. Since IK!,(0)1 > 1 for all v by Proposition 1.1, the sequence is also monotone increasing by (1). Hence a := 1im114,(0)1 n,(0) 1 exists. Since a 0 (indeed, a> 1), it follows that lim14(0)1= 1 (cf. 1.1.1). • Koebe, in 1915. called the algorithm a Sch.miegungsverfahren (osculation process). The nth osculation operation is the choice of the expansion K n , which produces the domain Gn +i from G (tKoe 4 1, pp. 183-185). The osculation effect is expressed by (2). Equation (3) is the key to forcing the desired equation lim r(h,(G)) = 1 by means of "cleverly" chosen sequences K n ; see Subsection 3.
2. Convergence theorem. We seek expansion sequences h, : G E that converge to biholomorphic maps G E. The following convergence theorem shows when this occurs. An expansion sequence h, : G —4 E is called an osculation sequence if h(0) is always positive and lirn r(h„(G)) =1. Convergence theorem. 1) Let h„ : G E be an expansion sequence that converges compactly to a function h. Then h: G E is an expansion, and
r(h(G)) > lim r(h„(G)). 2) Every osculation sequence h, : G —+E converges compactly in G to a biholomorphic expansion h : G —4 — E. Proof. ad 1). By 1(1), Ih(z)1 > Ihn(z)1 > 1.z1 for all Z E CVO} and all n E N. Since h(0) = 0, it follows that h is not constant; hence, since all the h„ are injections, h is injective by Hurwitz. Therefore h : G E is an expansion of G. Furthermore, it follows from the expansion lemma 1.1 that r(h(G)) lim r (h„ (G)) for all n. Hence r(h(G)) > lim r (h.„(G)) ad 2). Since subsequences of expansion sequences are again expansion sequences, 1) implies that every limit function h of a subsequence of the sequence h, is an expansion and hence maps G biholomorphically onto h(G) C E. Since moreover r(h(G)) > 1 by 1), we have h(G) = E; that is. h : G E is biholomorphic. To see that the sequence h„ converges compactly, it suffices since h„(G) c E to show that all its compactly convergent subsequences have the same limit (Monters convergence criterion 7.1.3). lf h, h are such limits, then, by what has already been proved, E with h(0) = h(0) = O. Since they furnish biholomorphic maps G h 1 (0) > > 0 for all n, it follows that h/(0) > 0, 711 (0) > 0: hence
h = h by the uniqueness theorem 8.2.5. Observe that 1(3) was not used in this proof and that the limit map in 2) is. by 8.2.5, the uniquely determined map G E. 0
196
8. The Riemann Mapping Theorem
Since simply connected domains C can be mapped biholomorphically onto Koebe domains (cf. 8.2.2), the Riemann mapping theorem follows from 2) once an osculation sequence h„ has been constructed for G. Special expansion families are used in this construction. In the next subsection we describe how Koebe constructed such a family. Since he wants limr(hr,(G)) = 1 and knows that lirn Ite„(0)1 = 1 (by 1(3)), he "artificially" produces a relationship between inner radii and derivatives of expansions. 3. Koebe families and Koebe sequences. 7- will denote a continuous real-valued function on (0, 1) for which T(x) > 1, x E (0, 1). An expansion family /C is called a Koebe family for T if (a)
K/ (0) = r(r(G))
for all
: G —> E
in /C.
E relative to a Koebe An expansion sequence h = Kn ok.„_10• • •Oko : G family K is called a Koebe sequence. If T is the function corresponding to K, it follows from (a) that
r(r(h,,(G))) = fren+1 (0),
(a*)
n E N.
From (a*) and 1(3), we now immediately obtain the crucial Osculation lemma. Every Koebe sequence is an osculation sequence. Proof. By 1(2), r := lim r(h r,(G)) E (0, 1]. If r were less than 1, the continuity of T in r, together with (a*), would imply that lim re„(0) = r(r). But this contradicts 1(3) since r(r) > 1. Thus r = 1. Since 11,',(0) > 0, 1(1) implies that len (0) > 0 for all n; hence 14, is an osculation sequence. C
To complete our study of Carathéodory-Koebe theory, the "only" thing we need to prove is the existence of Koebe families. For every Koebe domain G, we use the square root method of 1.2 to construct all expansions it G —+ E with fre(0) > 0, where c E E is always chosen in such a way that ca is a boundary point of G with minimum distance to the origin. We let G run through all Koebe domains and denote by K2 the set of all expansions obtained in this way. We set 1 — X -1 r2
2 xi
1+ X
= 2 ■/7
,
x E (0,1);
this function is continuous in (0, 1), and 72(x) > 1 for all x E
(0.1).
Proposition. The family K 2 is a Koebe family for r2 . Proof. By Theorem 1.2, K2 is an expansion family. For all ic E K2, K I M= 7-2 (r(G)) by 1.2(2), since W(0) > 0 and Ici 2 = r(G) by the choice of c.
§2. The Carathéodory-Koebe Algorithm
197
Other Koebe families are constructed in Section 3. 4. Summary. Quality of convergence. Carathéodory-Koebe theory contains the following as a special case.
Koebe's main theorem.
For every Koebe domain G there exist Koebe
domains G, with G o = G, and expansions Kr, : G„ -+ E, with k n (G„) = G n+ i, such that the sequence K„OK„_10•••0K0 : G E converges compactly to a biholomorphze expansion G -■ E. Every expansion K„ can be explicitly constructed by a square root operation: n, E /C2-
By 1.2(4), r(G) < r(K(G)) for all expansions K. E AC 2. Since icr = c(G), it can even be proved that r(K(G)) >
\Fr
1+r
(V2(1+ r 2 ) + r -
1),
if
r :=
1.■
E /C2
UCa2 ], pp. 285-286; also [PS], Part IV, Problem 91). For numerical approximation of the Riemann mapping function h : G Z E, the square root method converges very slowly. Suppose that the inner radius r„ of the domain h, (G) is chosen as a measure of the quality of the nth approximation h,, = K„ 0 o • • • 0 ko. Ostrowski showed in 1929 ([0st], p. 174): There exists a constant M > 1- Min, n > 0.
0,
depending on r(G), such that r, >
5. Historical remarks. The competition between Carathéodory and Koebe. Square root maps occur early in Koebe's work (e.g. in 1907 in [Koed, P. 203 and p. 644, and also in 1909 in [Koe2], p. 209 and p. 216). But they were first used systematically by Carathéodory in 1912, in his recursive procedure for constructing the Riemann mapping function. He works explicitly with the function z
(1 + r2 )z - 2re i ° 2rz - (1 + r 2 )ei°
Ka l b p. 401):
ru ' . in the terminology of 8.2.3, this is the function c - ' 6 1»,. with c To prove that his sequence converges compactly. Carathéodory used Montel's theorem (loc. cit., pp. 376-378). In 1914, in the Schwarz Festschrift, he explained his method in detail; he formulated the convergence t heorem explicitly and proved it directly, without resorting to Montel ([C a i l, pp.
280-284). Koebe did not let Carathéodory's breakthrough rest in peace. He had dedicated his life to conformal maps and enriched the theory by a vast abundance of papers — from 1905 to 1909 alone he wrote more than 14 papers. some of them quite long — and could immediately (in 1912) let the approximating Riemann surfaces that had appeared in Carathéodory's
198
8. The Riemann Mapping Theorem
work "arise automatically." Stimulated by "Mr. C. Carathéodory's interesting work," he takes up his "earlier thoughts again" and "reveals a new elementary method of mapping the most general [schlicht] simply connected region onto the surface of the unit disc," which has "ideal perfection in more than one respect"; cf. [Koe 3 ], pp. 844-845. Here Koebe sketches the square root algorithm for the first time. He presents his "osculation process" in detail in 1914 ([Koe4 ], p. 182): Die [Konstruktion] der konformen Abbildung des gegebenen Bereichs auf das Innere des Einheitskreises werden wir durch imendlich viele Quadratwurzeloperationen bewirken, ... die wesentliche Eigenschaft der einzelnen dieser Operationen ist, eine Versthrkung der Anschmiegung der Begrenzungslinie des jeweilig abzubildenden Bereichs an die Peripherie des Einheitskreises, und zwar vom Innern her zu bewirken. (We will carry out the [construction] of the conformal map of the given region onto the interior of the unit disc by infinitely many square root operations; ... the crucial property of each of these operations ... is to improve the osculation of the boundary curve of the region to be mapped to the boundary of the unit disc and, more precisely, to do so by working outward from the interior.) Koebe argues geometrically — as did Carathéodory before him; thus twosheeted Riemann surfaces are used again. In 1916, at the 4th Scandindavian congress of mathematicians, Lindel6f gave a detailed lecture on Koebe's proof; cf. [Lin]. G. P6lya and G. Szeg6, in 1925, split Koebe's proof into nine problems ( [PS], Problems 88-96, Part IV, pp. 15-16); they prove the convergence theorem directly (without Monte!) — as. of course, does Koebe. Their arguments are free of Riemann surfaces and simpler than in the pioneering work of Carathéodory and Koebe.
§3.
The Koebe Families IC„, and 1C,c
Koehe noticed immediately (in 1912) that, in his osculation process, taking square roots "could readily be replaced by taking roots of higher orders or taking logarithms" ([Koe2], p. 845). We want to construct corresponding Koebe families. In t he construction, we use the fact that holomorphic functions that have no zeros in Q-domains always have holornorphic mth roots. m E N, and holomorphic logarithms. -- In what follows, the (large) faniilyE = If :E — ■ E holomorphic, 1 (0) = 0, f Aut E, f(E) is not a Koebe domain) plays an important role. G again always denotes a Koebe domain. We first generalize Lemma 8.2.3.
§3. The Koebe Families IC, and /Cc,.
199
1. A lemma. Let (p E E, and let k : G
IE be a holomorphic map such that 0 and ;, o # = id. Then K, is an admissible expansion of G.
•
Proof. Since cp Aut E, Schwarz implies that ko(w)I < lwl, w E Ex . Since cp 0 K = id, K is injective; moreover, izi = Icp(k(z))1 < Ii(z) I if K(z) 0 0, i.e. if z E G\{0}. Hence K is an expansion of G. If K(G) were equal to E, then 'p(E) = G would be a Koebe domain, contradicting (p E E. Thus k is admissible by 1.2(1).
The trick is now to track down those functions cp E E for which there exists a tt as in the lemma. We give two examples for which this is possible. Example 1. Let m E m > 2, and let c E Ex . We denote the map E -4 E, • zrn, by j,, and claim:
The map c m := gen o ,
(o)
(1)
= mcm -i
o 9, : E E is in E and 1c I 2 — 1
1-1
m
1c 12— — 1
(
-i Ec i ))
Ici ' e
—
ici -'
r _ i ci _„ t • n
Proof. E E since j„, Aut E and W,,,,(E) = E. Equation (1) follows (by the chain rule) since j(c) = mc' l and ga' (0) = 1/g ia (a) = lal 2 - 1, a E E. Example 2. Let c E Ex. We choose b E H such that c = e tb . The map
qb : H -4 E, z 1-4
z-b
:E
with
z ■—•
b -bz 1—z
is biholomorphic (a generalized Cayley map). The function e(z) := e i z maps H onto E\101. We claim: The map xe := 9c oEoqIT I :E-4 E is mE and
(2)
, (0) xc
2 c log
1c12
-
icl = 2 1
c
logic'
Ici Ici 1c1 -1.
Proof. Since E: : H E\{0} is not biholomorphic and x(E) = E\ {c} is not a Qdomain, we have x, E E. Equation (2) follows, since x(0) = and g(c) = 1/(1W - 1), El (b) = iC, WlY(0) = b - = logic1 2 .
In the next subsection, for the functions Wrn,c, Xe E E, we construct maps satisfying the hypotheses of the lemma.
K
2. The families K m and ICO.a . The functions (1) -rm (x) :=
1 x-x-' m = 2,3,...; mxi/rn - x - i/nt
r.c.o (x) :=
x -1 , x E (0, 1), 2logx
are continuous and map (0,1) to (1, oc). (Prove this.) For every Koebe domain, we construct admissible expansions K such that Ifre(0)1 = r,„(r(G)), 2 < m < Do.
The mth root method, m > 2. Let c be chosen in such a way that cm is a boundary point of G with minimum distance to the origin. Let y E 0(G) be the E is an admissible mth root of gen1G with v(0) = c. Then i := gc o y : G
200
8. The Riemann Mapping Theorem
expansion of G and (2)
1K 1 ( 0 )1 = Tm(r(G)).
Proof. K :G —.E is well defined since v(G) C E. We have (*)
h;(0) = g c (c) = 0
and
Or ,„, o
K=
id
(because jr„ o V = g c m 1G).
Example 1.1 shows that E E; hence K is an admissible expansion of G by Lemma 1. By (*), le,./ (0)1 = 1/1-14,,(0)1. Thus (2) follows from 1(1) since le" I
r(G).
The logarithm method. Let c be a boundary point of G with minimum distance to the origin, and let iv E 0(G) be a logarithm of gc 1G. Then b := v(0), is an admissible expansion of G and (3)
K
:= gb o v, with
1k'( 0 )1 = T.x, (r(G)).
Proof. Since e" = gc (z) E E, z E G. we have v(G) C ill; in particular, b E Hence K. : G —0E is well defined. It follows (with E o v = gc ) that (*)
K(0) = qb(b) = 0
and
x e o = id.
Example 1.2 shows that x, E E; hence K is an admissible expansion of G by Lemma I. By (*), I re (0) I = 1 /Ix( 0 )l. Thus (3) follows from 1(2) since Icl = r(G). 0
In addition, we "normalize" each expansion obtained above (by multiplying by an a E S I ) so that re(0) > 0; this normalized expansion is again admissible. Now let /C,,, and K oc, denote, respectively, the families of all normalized expansions constructed by means of the mth root method and the logarithm method. Then 2.3(a) is satisfied by T := 7,n and by T := Too . The next assertion follows. Proposition. JC„, is a Koebe family for r.,7„ in = 2, 3, ... ; AC for T.
is a Koebe family
Remark. The family /C,, is the "limit" of the families K m : For fixed G arid c E aG, each expansion K E 1C c,0 is the limit of a sequence k m E k m . For the functions r„,. and 7-00 , it is immediate that lim rn,(x) = r(x) for all x E (0, 11. For the proof of the Riemann mapping theorem, the square root method is generally used. Carathéodory applied the logarithm method in 1928, choosing c = h E (0, 1) and working with the function h — exp [(log h) (cf. [Ca l ], p. 304, formula (6.2)),
1 — hexp [(log h) which is precisely the function xh with b = —i log h. (Prove this.) This says, by the way, that he could just as well have chosen the function
z(2A — (1 + h)z) (1 -1- h) — 2N/Tiz
thus
1,b2,h
(cf. p. 305).
§3. The Koebe Families 1Cm and IC,
201
H. Cartan ([Car2], p. 191) also uses the logarithm method. He works with the admissible expansion R := gb o 1/, where eb = -c and V E 0(G) is a logarithm of -gc ; it still holds that RI (0) = -r o (r(G)). The reader should carry out the computations and determine akEE with )? o R = id. The families /Cm . ke.Q , and others are treated in detail by P. Henrici; see [He ] , pp. 328-345, where the rate of convergence of the corresponding Koebe algorithms is discussed. The functions h , nt. m > 2, h E (0, 1), and xh are studied by Carathéodory in [Ca2], pp. 30-31; it is also shown there that lim Oh o?, = Xn •
Bibliography for Chapter 8 and the Appendix [B]
BIEBERBACII. L.: Neuere Untersuchungen über Funktionen von komplexen Variablen, Encyc. Math. Wiss. II, 3.1, 379 532, Teubner, 1921. CARATHEODORY, C.: Ges. Math. Schriften 3.
ra21
CARATHgODORY, C.: Conformal Representation, Cambridge University Press 1932, 2nd ed., 1952. CARTAN, E.: Sur les domaines bornés homogènes de l'espace de n variables complexes, Abh. Math. Sem. Hamburg 11, 116 162 (1935); (Eitvres I, 2, 1259-1306.
[Card
[Car2]
CARTAN, H.: Elementare Theorze der analytischen Funktionen eviler oder mehrerer komplexer Veriinderlichen, Hochschultaschenbücher BI, Mannheim,
1966. [COU1
COURANT, R.: Über die Anwendung des Dirichletschen Prinzipes a.tif die Probleme der konformen Abbildung, Math. Ann. 71, 145-183 (1912).
[Fiel
HENRIC:I, P.: Applied and Computational Complex Analysis, vol. 3, J. Wiley and Sons, 1986.
[Hi]
HILBERT, D.: Ces. Math. Abh. 3, Springer, 1970.
(Ka ]
KAUP, L and B. KALIP: Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory. trans. M. BRIDCLAND, de Gruyter, 1983. KOEBE, P.: Ueber die Uniformisierung beliebiger analytischer Kurven. Nachr. Köntgl. Ces. Wiss. Gottingen, Math.-phys. Ki. 1907, erste Mitteilung 191-210; zweite Mitteilung 633 -669. KOEBE, P.: Ueber die Uniformisierung beliebiger analytischer Kurven 1, Math. Ann. 67. 145-224 (1909). KOEBE. P.: Ueber eine neue Methode der konformen Abbildung und Uniformisierung, Nachr. K6itzgl. Ces. Wiss. Göttingen, Math.-phys. Kt. 1912, 844-848. KOEBE, P.: Abliandlungen zur Theorie der konformen Abbildung, I, Die Kreisabbildung des allgemeinsten einfach und zweifach zusammenhngenden schlichten Bereichs und die Rinderzuordnung bei konformer Abbildung, Journ. reine angew. Math. 145, 177-223 (1915).
iKoe i ]
11Coe21 [Koe3]
[Koe4 ]
202
8
The Riemann Mapping Theorem
[Lich]
LICHTENSTEIN, L.: Neuere Entwicklungen der Potentialtheorie, Konforme Abbildung, Encykl. Math. Wiss, H 3.1, 177-377, Teubner, 1919.
[Lin]
LINDEL6F, E.: Sur la représentation conforme d'une aire simplement connexe sur l'aire d'un cercle, Compte rendu du quatrième congrès des mathématiciens scandinaves, 59-90; ed. G. MITTAG-LEFFLER, 1920.
[N]
NARASIMHAN, R.: Complex Analysis in One Variable, Birklauser, 1985.
[Osg]
OsGoot), W. F.: On the existence of the Green's function for the most general simply connected plane region, Trans. Amer. Math. Soc. 1, 310314 (1900).
[Ost]
OsTRowsKi, A.: Mathematische Miszellen XV. Zur konformen Abbildung einfach zusammenMngender Gebiete, Jber. DMV 38, 168-182 (1929); Coll. Math. Papers 6, 15-29.
[Po]
POINCARI:], H.: Sur les groupes des équations linéaires, Acta Math. 4, 201311 (1884); Œuvres 2, 300-401.
[Porn]
PomMERENKE, C.: Boundary Behaviour of Conformal Maps, Springer, 1991.
[PS]
P6LYA, G. and G. SzEG6: Problems and Theorems in Analysis, vol. 2, trans. C. E. BILLIGHEIMER, Springer. New York, 1976.
[Ra]
RAD(5, T.: Ober die Fundamentalabbildung schlichter Gebiete, Acta Sci. Math. Szeged 1, 240-251(1922-1923); see also Fejér's Ges. Arb. 2. 841-842.
[Ran]
RANGE, R. M.: Holomorphic Functions and Integral Representations in Several Complex Variables, Springer, 1986.
[R le]
RIEMANN, B.: Grundlagen fiür eine allgemeine Theorie der Functionen einer veriinderlichen complexen Grosse, Inauguraldissertation Giittingen 1851, 147erke 3-45.
[Schw]
SCHWARZ, H. A.: Ces. Math. Abhandlungen, 2 vols., Julius Springer, 1890.
[Wei]
WEIERSTRASS, K.: Ober das sogenannte Dirichletsche Princip, Math. Werke 2. 49 54.
9 Automorphisms and Finite Inner Maps
The group Ant G and the semigroup Ho! G, which were already studied in 8.4, are central to Sections 1 and 2. For bounded domains G, every sequence E Ho! G has a convergent subsequence (Montel); this fact has surprising consequences. For example, in H. Cartan's theorem, one can read off from the convergence behavior of the sequence of iterates of a map f : G G whether f is an automorphisrn of G. In 2.5, as an application of Cartan's theorem, we give a homological characterization of automorphisms. An obvious generalization of the biholomorphic maps G G' is the finite holomorphic maps G —4 G', for which all fibers are finite sets which always have the same number of points (branched coverings). Such maps are studied in Sections 3 and 4; we show, among other things, that every finite holomorphic map of a nondegenerate annulus onto itself is linear (fractional), hence biholomorphic.
§1.
Inner Maps and Automorphisms
We first show that composing inner maps and taking inverses of automorphisms are compatible with compact convergence (Subsection 1). The proofs are easy, since the sequences fri E 0(G) that converge compactly in G are precisely those that converge continuously in G to their limit f, and thus for which lim f n (c„) = AO
if
(see the footnote on p. 150 or 1.3.1.5*).
lim c,, = cE G
204
9. Automorphisms and Finite Inner Maps
We write f = lim f„, if the sequence f„ e 0(G) converges compactly (continuously) to f E 0(G). If G is bounded and fn E Hol G, then, by Vitali, f = lim fn if the sequence f converges pointwzse to f. 1. Convergent sequences in Hol
G
and Aut G. The next statement
follows immediately from 7.5(1). (1) If frt E Hol G and lirn f = f E 0(G) is not constant, then f E Bol G. We also note:
(2) If f„ E Hol G, g,, E 0(G) and limf = f E Hol G, lim g,„ = g E 0(G), then lim(g„ o f„) =gof E 0(G). Proof. Let c„ E G be a sequence with limit c E G. Continuous convergence implies that inn fn(c,1) = f (c) and also that lirn g,1 (fi, (cn)) = g( f (c)). The sequence g„ o f,, therefore converges continuously in G to g o f. For the inverse map. we show:
(3) If fi, E Alit G and lim fn =- f E AutG, then lim
= f' E Aut G.
Proof. Let c„ E G be a sequence with limit c E G. For b„ f,-, 1 (e„) C G. b := f -1 (c) E G, we then have f7 (b) = f (b). It follows from 7.5(2') = f -1 ) and hence that the sequence frr 1 converges conthat lim fn- 1 ( tinuously to f -1 . Remark. Behind (2) and (3) lies a general theorem on topologizing transformation groups. If X is a locally compact space, then the set of all homeomorphisms from X onto itself is a group Top X. Consider the sets If E Top X : f (K) C U1, where K is compact and U is open in X. The family of all finite intersections of such sets forms a base for a topology on Top X. In this so-called C-0 topology (compactopen topology), the group operation Top X x Top X Top X, (f , g) f o g, is continuous. If X is moreover locally connected, the inverse map Top X -* Top X. f f -1 •is also continuous. Hence the following holds: Theorem (Arens [A], 1946). If X is locally compact and locally cdnnected, then Top X — equipped with the C-0 topology — is a topological group.
In the case of domains G in C, Aut G is a closed subgroup of Top G: the sequences that converge in the C-0 topology are precisely those that converge compactly in G. If G is bounded, the topological group Aut G is locally compact and in fact a Lie group. These results were proved in 1932 and 1936 by H. Cartan for all bounded domains in C", 1 0, and outer radii s, s' < cc. In order to study finite maps A —, A' we need a purely topological lemma, which generalizes the boundary rule 2(1) and, intuitively, says that boundary components of A are mapped to boundary components of A', where the inner and outer boundaries may be interchanged. Boundary lemma. If f : A —, A' is holomorphic and finite, then either
lim If (z)I = r' 121—'7'
and
lirn 1f(z)1 = s'
izl's
214
9. Automorpliisms and Finite Inner Maps
or
lini I f (z)I = .s'
and
fim I f (z)1 -=
.
Proof. Let t E (r'. s') be fixed, and let S := C A'. Since f E C: Izi = is finite, f -1 (S) is compact in A by 1(2) and hence has a positive distance from OA. Thus there exist numbers p, a with r V is a winding map of degree n about e, then for every disc C V about f(e) the induced map : U, -4 VI. where U 1 := f 1 (V1 ), is a winding map of degree n about e. Prvol. Let ropo u, with p(z) := zn , be a factorization of f as in a) and b). Since t , is linear, B' := y -1 (V1 ) is a disc about O. If r is its radius, then B := p-1 (B') is tlw disc about 0 of radius s := VT. If we now set u (z) := s it(Z), Z E U1, and e l (2) := z E E. then y 1 o p o u1 gives a factorization of f 11 1 as a winding map. The v( f. c) of f at c E G is the order of the zero of f — [(c) at c. For nonconst ant f , tlw inequality 1 < v(f, c) < oc always holds: v(f. c) = 1 if and only if f is biholomorphic in a neighborhood of c; that is, if f' (e) O. For more on this. see 1.8.1.4 and 1.9.4.2.
§4*. Rad6's Theorem. Mapping Degree
219
We now generalize (1). Let f E 0(G) have finite fibers, and let c i , be the distinct points of such a fiber f - ' (p). Then f has the following representation about f -1 (p): Proposition. There exist an open disc V about p and open patrons(' disC G of , r„, such that V = f (t joint neighborhoods flUn : (In V is a winding map about r of degree and every 'induced map v(f,c n ). 1 < p < m. f is finite, the sets Up and V can be chosen so that f -1 (V) 1 11 U (12 U • • • U Um.
0„,
Proof. Using (1). we choose pairwise disjoint 01 c G about e 1 . f (Up ) is a winding map about c of degree cm such that f 'Up U about p. If we set v(f.e n ),1< p < m. There exists a disc V C •(
o„ )
:= f -1 (V) n 0,1 then. by (2). flUn 4 V is also a winding map about e of degree v( f , en ), 1 < p < m. If f is finite, then f is closed: hence, by 1(1) and the shrinkage property (2), V can be chosen so that f -1 (V) = Ul l.J• • • U U„,. -
,
Corollary. If f : G —> G' is finite and locally biholomorphic. then every point p E G' has a neighborhood V for which fl (V) can be decomposed into finitely many domains U .) , j E J, in such a way that every induced map f :Uj —> V is bih,olomorph,ic. Such maps are also called finite-sheeted (unbranehed) coverings. 3. Rad6's theorem. If f E 0(G) is nonconst ant, then for every point w E C the number of points in the fiber f -1 (w) is measured by degy ,f :=
E
v(f. e)
if w G f (G),
deg w f :=
otherwise.
(,(i-1(1D) We have the equivalence the fiber f -1 (w) is nonempty and finite.
1 < deg w f < oc
1. deg w q = d for all w E C (fundamental theorem of algebra). For winding maps f : G 4 G' about c E G. the degree function is also constant: deg„.f = v(f,e) for all w E G'. We prove tlw following general result. -
Theorem (Rad6). A holomorphic map f G —> G' is finite if and only if its degree function deg,„f is finite and constant ni G.
220
9. Automorphisms and Finite Inner Maps
Proof. For every point p E G' with 1 < degpf < co, we choose V, U,,. . . as in Proposition 2. We set U := U1 U • • U Urn . Since U1 Um are pairwise disjoint, it follows from Proposition 2 that =
(1)
deg„,( f IU) =
E degw (
=E v( f , p=3.
1 ,1
= degp f.
w e V.
We can moreover arrange that U is compact and lies in G. 1. Let f be finite. Let p E G' be arbitrary. By Proposition 2, we may assume that U = f-1 (V). Then degw f = deg w (fiU) for all w E V; hence deg„, f = degp f for all w E V by (1). Thus the degree function deg„, f is locally constant, and hence constant in G'. 2. Let degw f be finite and constant in C'. If f were not finite, there would exist a boundary sequence z„ in G whose image f(z) had a limit p E G'. By ( l), it would follow that deg. f=degw (f 1U) for all w E V. Hence f -1 (w) G U for all w E V. Since p = lim f (z n ), it thus follows that z„ E U for almost all n. But this is impossible because U C G is compact and zn is a boundary sequence in G.
Rad6 proved this theorem in 1922 ([N, pp. 57-58). The theorem and proof hold verbatim for arbitrary Riemann surfaces G, G'.
4. Mapping degree. For every finite map f : G G', deg f
degw f =
E
v(f, c), w G G'
rEf -1 (w)
is a positive integer by Theorem 3; it is called the mapping degree of f. Polynomials of degree d define finite maps C C of mapping degree d. The integers d > 1 that appeared in the theorems of Subsection 3 are always the degree of the corresponding finite maps. We emphasize: (1) The finite maps f : G maps.
G' of degree I are exactly the biholomorphic
For every function f E 0(G), the set 5 := {z E G : (z) = 0} is called the branch locus of f. If f is nonconstant, then S is locally finite in G. For finite maps f : G G', f (S) is therefore always locally finite in G'. The following is immediate from Rad6's theorem. (2) If f : G —* G' is finite with degree d, then every fiber has at most d distinct points. The fibers over G"\f (S) are exactly those that have d distinct points. (The induced map G\f -1 (f(S)) —+ G\f(S) is a d-sheeted covering.)
The following is another immediate consequence of Rad6's theorem. Degree theorem. If f : G G' and g : G" are holontorphic and finite, then gof :G G" is also finite, and deg (g o f) = (deg g) • (deg f).
§4*. Rad6's Theorem. Mapping Degree
221
This statement is to be viewed as an analogue of the degree theorem
[M : K] = [M : 1,][1, : K] of field theory (M is an extension field of L and L is an extension field of K). For lack of space we cannot pursue the interesting connections further; see the exercise.
Historical note. Rad6 introduced the general concept of finite holomorphic maps in 1922 [R]; he called them (1. m) conformal maps, where m is the degree. Exercise. Let f : G
G' be holomorphic arid finite. Regard 0(G) as a ring containing 0(G') (with respect to the lifting f - : 0(G') 0(G), h h (.) f ) and prove: a) For every g E 0(G). there exists a polynomial w(Z) = Z" + an E 0(G')[Z], with n = deg f such that w(g).= 0. b) If g is bounded, then a l , a„ are also bounded. C. c) If G is bounded, then G'
Z" -1 + • +
5. Glimpses. That f is holomorphic is used only superficially in the proof of t he equivalences in 1. (To prove the implication iii) i), one needs only that f • as a nonconstant function. is nowhere locally constant.) The theorem can therefore he generalized. We sketch a more general situation. Let X and Y be metrizable locally compact spaces whose topologies have countable bases. A continuous map f : X Y is called proper if every compact set in Y has a compact preintage under f One can prove the following: a) A continuous map X Y is proper if and only if it maps boundary sequences in X to boundary sequences m Y. b) Every proper map is closed.
Finite maps are now defined as proper maps whose fibers are all discrete sets; this definition is equivalent to ours in the holomorphic case Finite holomorphic maps play an important role in the function theory of several variables. With their aid, the n-dimensional local theory can be developed in a particularly elegant way; the reader can find this carried out systematically in [GR.], Chapters 2 3. All proper holomorphic maps between domains in C" , 1 < 7? < are automatically finite. The situation changes if one studies maps between arbitrary complex spaces: since holomorphic maps can now be dimension-lowering (without being constant), there exist many proper (hut not finite) holomorphic maps. For all such maps. Grauert's famous coherence theorem for image sheaves is valid; cf. IGR), Chapter 10, especially p. 207.
Bibliography [Al
ARENS, R.: Topologies for
homeomorphism groups, Amer. Journ. Math.
68, 593-610 (1946). BIEBERBACII. L : fiber einen Satz des Herrn Carathéodory, Nachr. Ktintgl. Gesellschaft Wiss. Göttingen, Math.-phys. Ki.. 552-560 (1913). CARTAN, H.: Œuvres 1, Springer, 1979.
222
9. Automorphisms and Finite Inner Maps
[F1]
FATota, P.: Sur les équations fonctionelles, Bull. Soc. Math. Prance 48, 208-314 (1920).
[F2]
FATOU, P.: Sur les fonctions holomorphes et bornées à l'intérieur d'un cercle, Bull. Soc. Math. France 51, 191-202 (1923).
[GR]
GRAUERT, H. and R. REMMERT: Coherent Analytic Sheaves, Grdl. math. Wiss. 265, Springer, 1984.
[Heil]
HEINS, M.H .: A note on a theorem of Rad6 concerning the (1, m) conformal maps of a multiply-connected region into itself, Bull. Amer. Math. Soc. 47, 128-130 (1941).
[Hei2]
HEINS, M.H.: On the number of 1-1 directly conformal maps which a multiply-connected plane region of finite connectivity p( > 2) admits onto itself, Bull. Amer. Math. Soc. 52, 454-457 (1946).
[Hu]
HUBER, H.: Über analytische Abbildungen von Ringgebieten in Ringgebiete, Comp. Math. 9, 161-168 (1951).
[K]
KOEBE, P.: Abhandlungen zur Theorie der konformen Abbildung, I. Die Kreisabbildung des allgemeinsten einfach und zweifach zusammeniangenden schlichten Bereichs und die Riinderzuordnung bei konformer Abbildung, Journ. reine angew. Math. 145, 177-223 (1915).
[R]
RAD6, T.: Zur Theorie der mehrdeutigen konformen Abbildungen, Acta
Litt. Sci. Szeged 1, 55-64 (1922-23).
Part C Selecta
10 The Theorems of Bloch, Picard, and Schottky
Une fonction entière, qui ne devient jamais ni à a ni 6, b est nécessairement une constante. (An entire function which is never equal to either a or b must be constant.) — E. Picard, 1879
The sine function assumes every complex number as a value; the exponential function omits only the value O. These examples are significant for the value behavior of entire functions. A famous theorem of E. Picard says that every nonconstant entire function omits at most one value. This so-called little Picard theorem is an astonishing generalization of the theorems of Liouville and Casorati-Weierstrass. The starting point of this chapter is a theorem of A. Bloch, which deals with the "size of image domains" f (1E) under holornorphic maps; this theorem is discussed in detail in Section 1. In Section 2 we obtain Picard's little theorem with the aid of Bloch's theorem and a lemma of Landau. In Section 3 we introduce a classical theorem of Schottky, which leads to significantly sharpened forms of Picard's little theorem and the theorems of Montel and Vitali. The short proof of Picard's great theorem by means of Mantel's theorem is given in Section 4.
The Theorems of Bloch,
226
10.
§1.
Bloch's Theorem
Picard, and Schottky
One of the queerest things in mathematics. ... the proof itself is crazy. -- J. E. Littlewood
For every region D C C, let 0(D) denote the set of all functions that are holomorphic in an open neighborhood of D = D U OD. Bloch's theorem. If f E 0(g) and f'(0) = 1, then the image domain f(E) contains discs of radius 2 — N/ > 12 • The queer thing about this statement is that for a "large family" of functions a universal statement is made about the "size of the image domains." In our proof of this in Subsection 2, we give a center for a disc of the asserted radius. (That the point f(0) is in general not such a center is shown by the function f, (z) := (e nz _ 1)In = z + • • • , which omits t he value —11n.) Corollary. If f is holomorphic in the domain G c C and f l (c) at a point c E G, then f (G) contains discs of every radius ..sif'(c)1. 0 G s < d(c , OG).
Proof. We may assume c = O. For 0 < s < d(c, OG), we have /3,9 (0) c G; hence g(z) := f (s z) s PO) E 0(E). Since g' (0) = 1, Bloch's theorem implies that g(E) contains discs of radius 1/12. Since f (B,(0)) = the assertion follows. In particular, the corollary contains the following: If f E 0(C) ts nonconstant, then f(C) contains discs of every radius.
3-
For connoisseurs of estimates, the bound 0.0858 will be improved in Subsection 3 to — 2 P, 0.1213 and even. in Subsection 4, to 0/4 0.43301. The optimal value is unknown: see Subsection 5. — For the applications of Bloch's theorem in Sections 2, 3, and 4, any (poor) bound > 0 suffices. 1. Preparation for the proof. If G C C is a domain and f E 0(G) is nonconstant, then by the open mapping theorem 1(G) is again a domain. There is an obvious criterion for the size of discs in the image domain.
(1) Let G be bounded, f G
C open. C continuous, and fr : G Let a E G be a point such that s := tninzEacif (z) — f (a)1 > O. Then f(G) the disc B„(f (a)). contais
Proof. Since Of(G) is compact, there exists a point. w* G 3 1(G) such that d(0 f (G), f (a)) = w, — f(a)1. Since G is compact, there exists a sequence z, e G with lirn f(z,) = tv,. and z, := lim z, E G. It follows that
§1. Bloch's Theorem
f(z.)= w. E Of(G). Since JIG is open, z, cannot lie in G. Hence and therefore 1w 1 — f (a)I s. It follows that B f (a)) C f (G).
z.
227 E
aG
We apply (1) to holomorphic functions f. The number s certainly depends on (a) and f ia (example: f(z) = ez in E). For discs V := r>0, there are good estimates from below for s.
Iry
Proof. We may assume that a = f (a) = O. Set A(z) := f (z) — f' (0)z; then A(z) = f [fl(C)— f' (0)14', [0,zi
whence
1A(z)1
f
If' (zt) — f'
IzIcit.
For y E V, Cauchy's integral formula and standard estimates give
v fi (11) ft
27ri
lvi
(C)c1(
Jay ((( — v)
1
Inv)
(0 )1
If Iv
.
It follows that
(*
1 Iztl
)
Iry
1 Iz1 2 , 27. -121 1f Iv.
IA(z)I < jo r — Izt1 IzIdt
Now let p c (0,r). The inequality If (z) f i (°) z i If / (°) IP — If (z)I holds for z such that IzI-= p. Since If 'Iv it follows from (*) that 21.f'(0)1,
\ If(Z)1 It( 0)1. Now p — p 2 1 (r — p) assumes its maximum value, (3 — 2N5)r, at p1 := (1 — E (0,r). It follows that If(z)I — 2Arlf 1 (0)1 = R for all 1z1 = p1 . Setting G := Bp. (0) in (I) shows that BR(0) C 1(G) C f (V). 0 Exert-3.5e. Use (1) to show that, for all f G 0(E) with f(0) = 0 and t(0) = 1, the
inclusion f (s) D BT (0) holds with r := 1/6 If I.
2. Proof of Bloch's theorem. To every function f E 0(E), let us assign the function If' (z)1(1 — 1z1), which is continuous on E. It assumes its maximum M at a point p E E. Bloch's theorem is contained in the following Theorem. If f E 0(E) is nonconstant, then f(E) contains the disc about — > (0)1. PP) of radius
228
10. The Theorems of Bloch, Picard, and Schottky
Proof. With t := -1(1 — 11)1), we have
M = 2 t1P(P)1 , Bt(P) C E, 1 —
t
for
z E Bt(P)•
From f'(z)1(1 — 1z1) 5_ 2t1f(P)I, it follows that Ir(z)1 2 1P(P)I for all z E 13 t (p). Hence, by Lemma 1, BR(f(p)) c (E) for R (3-2v)tir(p)i.
Historical note. A. Bloch discovered this theorem in 1924 (in fact in a sharper form; cf. Subsection 4); see [Bld, p. 2051, and [B12]. G. Valiron and E. Landau simplified Bloch's arguments considerably; see [L2], which contains a "three-line proof in telegraphic style - . Landau reports on the early history of the theorem in [L3 ]. The proof given above is due to T. Estermann ([E], 1971). It is more natural than Landau's proof ([L4 ] , pp. 99-101) and, for those who like bounds, yields"I — > which is better than Landau's .11Ti ; for more on this. see Subsection 5. A theorem of the Bloch type was first proved in 1904 by Hurwitz. He used methods from the theory of elliptic modular functions to prove (Math. Werke 1, Satz IV, p. 602): If f E 0(E) satisfies f (0) = 0. f' (0) = 1, and f(EX) C C x , then
f(E) D B,(0)
for s>
= 0.01724.
Carath6odory showed in 1907 (Ges. Math. Schriften 3, pp. 6-9) that in Hurwitz's situation 1/16 (rather than 1/58) is the best possible bound.
3*. Improvement of the bound by the solution of an extremal problem. In Theorem 2. the auxiliary function If ' (z)I(I - 1z1) was introduced without motivation. Here we discuss a variant whose proof is more transparent: the other auxiliary function Ir(z)1( 1 Iz12) and the better bound — 2> -112- \/ appear automatically.
Let f E 0(g) be nonconstant. The hope that f (E) contains larger discs as (0)I increases leads to an Extremal problem. Find a function F E 0(e) such that F(E) = f (E) and F has the greatest possible derivative at 0 (extremal function). To make this precise we consider, for a given f, the family
:={h= f j, j E Aut E}, (1)
where
j(z) :=
EZ - Lt1
•
E
71)
E E.
17VEZ -
Since j E 0(g) and j'(0) = e(1w1 2 — 1), it is clear that
(2)
h E 0(E), h(E) = f(E),
111/ (0)1 = It(w)1(1 — lw12)
§1. Bloch's Theorem
229
for every h= foj E F. Since f' E 0(E), the (auxiliary) function on the right-hand side assumes its maximum N > 0 at a point q E E. Thus one solution of the extrema1 problem is the function
(3)
F(z) := f (_z
E 0(1-E),
F(0) = f (q)
where
and
= .maxiw15.1 ir (w)1( 1 – 1 111 1 2 ). The following estimate for the derivative of F is now crucial: IF/ (z)i
(4)
1_1z1 2
maxi .1 „12contains discs of radius Proof. Choose F as in (3). Since IP(0)1 = N and, by (4), 1P(z)1 5. N/(1– Iz1 2 ), it follows that IF'(z)I < 21P(0)1 for all 1.z1 < By Lemma 1, f(E) = F(E) therefore contains the disc about f (q) = F(0) of radius
(3N7 – 2)N. The extremal problem has led us to the auxiliary function If '(z)j(1 –1 z 1 2 ) . It is clear that M < N; thus the new lower bound is obviously better than that in Theorem 2. Remark. The auxiliary function Ift (z)1(1 – Iz1 2 ) and its maximum N in E were introduced in 1929 by Landau ([L3], p. 83). All functions in the set
B := {f E 0(E) : sup It(z)1(1 –1z1 2 )
C is holomorphic, a disc B c f(G) is called schlicht (with respect to f) if there exists a domain G* C G that is mapped biholomorphically onto B by f. Ahlfors's theorem. Let f E 0(t); set N := maxkl 0 for all z E G. Then h maps G biholomorphically onto h(G).
Proof. For u, y E G the line segment -y(t) = u + (v - u)t, 0 < t < I. lies in G. Hence h(v) - h(u) = (v - u)[1. Re h'e-y(t))dt + f lin
(-y(t))dt]
0
if u y, since the first integral on the right-hand side is positive because Re h'(z) > 0. We now prove the theorem. We may assume that N = 1. Choose F as in 3*(3); then F'(0) = E S l . First let q = 1. By the lemma, Re r(z) > in B := Bp (0), p := 1/0: hence FIB : B -+ F(B) is biholomorphic by a). For all ( = pri`P E 0B, the lemma implies that
IF(() -
F(0)1 = fo p (te )dt > f Re F1 (tez)dt
>
11 '41
Jo
(1 - 11t)3
dt = -4-V3 .
Hence, by 1(1), F(B) contains discs of radius 0/4. Since f =Fog with g E Aut E, f maps the domain G := 9-1 (B) c E biholontorphically onto a domain G* that contains discs of radius 0/4.
§1. Bloch's Theorem
231
For arbitrary ri E S 1 , one works with rr i F. Then f : G —■ riG* is biholomorphic and nG* like G*, contains schlicht discs of radius 0/4.
We now come to the proof of the lemma. We first observe:
It suffices to prove the estimate (*) for all real z E [0,1/ A. F(elP z) satisfies the hypotheses of the lemma for Indeed, Fcp (z) := every ço E IR. Thus (*) holds for F and z = re i s° if it holds for F and z = r. The proof itself is rather technical and requires:
b) For z := p(w) := 0(1 — w)/(3 — w) and q(w) := "lw(1 — followmg statements hold: p(E) C E; q(p -1 (z)) =
1
—vz 3;
the
p maps [0,1] onto [0, -A-];
and
lq(w)I(1 — ip(w)1 2 ) = 1 for all w E 0E.
.z) (1—
c) If h E 0(E) with h'(0) = 1 and 1h' (z)1 < 1/(1 — lz1 2 ), then h"(0) = 0. The proof of b) is a routine calculation. the antiderivative
L
For the proof of e), consider
WM< = z + az 2 + • • e 0(E)
of h'. For all z = re' G E,
«r iz + az2 + 0
t)lcil Ie"P hf(
r dt fo 1— It1 2 =
1 + 3 Iz1 3 4- • • • •
Since the quadratic term in lz I is missing on the right-hand side, considering small zI shows that a = 0 and hence that h"(0) = 2a = O. After these preliminaries, the proof of (*) in the lemma goes as follows. Consider the auxiliary function (P(w)) q(w)
H(w) :=
1
(1 — w) 2
(w 1(1 — w) 2 is Koebe's extremal function). Since p(E) c E, H is holomorphic everywhere in 1E \111. Since F"(0) = 0 by c), H is also holomorphic at 1 E C. It follows that H E 0(1g). Now w/(1 — w) 2 is real and negative (< —i) for all w E aEvil. Hence Re H (w) =
(1 — w)
Re
( F' (P(w)) 1)
q(w)
for all
W
E 0E.
232
10. The Theorems of Bloch, Picard, and Schottky
The inequality 1r(z)1(1 - 121 2 ) < 1 and the last equation in h) imply that IF1 (p(w))1 < lq(w) I on aE. Since Re(a - 1) < 0 for every a E E, it follows that Re H(w) > 0 for all w E aE. Applying the maximum principle to e -11( " ) now gives Re H(w) > 0 for all w E E. Since q(w) > 0 and w/(1 - w) 2 > 0 for w E [0, 1), it also follows that
Re F' (p(w))
g(w)
for all w E [0, 1].
By the statements in b), this is (*) for all z E [0,1/01.
The next result follows immediately from Ahlfors's theorem (see the introduction): If f E 0(C) is nonconstant, then f(C) contains arbitrarily large schlicht discs.
5*. Landau's universal
constant. Bloch's theorem and its variation
prompted Landau to introduce "universal constants"; cf. [L3], pp. 609615, and [I ], p. 149. For every h E F := {f E 0(E) : P(0) = 11, let Lh denote the radius of the largest disc that lies in h(E) and let Bh denote the radius of the largest disc that is the biholomorphic image under h of a subdomain of E. Then 44
L :=inf{Lh : h E .F}
and
B :=inf{Bh: h E Y.}
are called Landau's and Bloch 's constants, respectively. Landau correspondingly defines the numbers Ah and A for the family .F* := 1h E F : h injective}. It is trivial that B < L < A. Only bounds are known for B, L, and A; thus, in the preceding subsection, we showed first that L > 3 - ;■-', 0.0858 and then that L > 2 0.1213. Ahlfors's theorem even says that B > PA.,- 0.4330. In [Bon], Bonk shows more: B> N/F3+ 10 -14 . Since log ei E .F*, certainly A< 7-14 -7r 0.7853. Thus -
0.4330 + 10 -14 < B < L < A < 0.7853. Such estimates and refinements continue to fascinate function theorists; it has been proved (cf. [L4], [M], and [Bon]) that
0.5 < L < 0.544, 0.433 + 10 - ' 4 < B < 0.472, 0.5 < A. More recently, Yanagihara [Y] proved that 0.5 + 10 holds that B < L < A. The
Ahlfors-Grunsky conjecture: B=
2-
has been unsolved since 1936.
1 F(
'
)11(14)
F()
- 0 4719
< L. It actually
§2. Picard's Little Theorem
233
The theorem of Hurwitz and Carathéodory also has a corresponding universal constant. "Diese m6chte ich aber nicht die Carathéodory Konstant C nennen, da hell Herr Carath(fiodory festgestellt hat. da8 sic schon einen anderen Namen, 1/16, hatte." (But 1 would not like to call it the Carathéodory constant C, as Mr. Carathéodory determined that it already had another name, namely 1/16.) ([1.3 1. p. 78)
§2.
Picard's Little Theorem
Nonconstant polynomials assume all complex numbers as values. In contrast, nonconstant entire functions can omit values, as the nonvanishing exponential function shows. With the aid of Bloch's theorem, we will prove: Theorem (Picard's little theorem). Every nonconstant entire function omits at most one complex number as value. This statement can also be formulated as follows: Let f E 0(0) and suppose 0 1 (C) and 1
f (C) Then f is constant.
The theorem follows immediately from this: Indeed, suppose h E 0(0) h; then [h(z) — a11(1) — a.) E 0(C) omits omits the values a, b, where a the values 0 and 1 and is therefore constant. Hence h is also constant. 0 In C, meromorphic functions can omit two values: for instance, 1/(1+e') is never 0 or 1. This example is significant in view of Picard's little theorem for meromorphic functions. Every function h G M(C) that omits three distinct values a,b,c E C is constant. The function 1/(h — a) is then entire and omits 1/(b — a) and 1/(c — a). In Subsections 1 and 2, we give the Landau-K6nig proof of Picard's little theorem (cf. [L4 ] , pp. 100 102, and [K]).
1. Representation of functions that omit two values. If G C C is simply connected, then the units of 0(G) have logarithms and square roots in 0(G); cf. 8.2.6. Our next result follows just from this, by elementary manipulations. Lemma. Let G C C be simply connected, and let f E 0(G) be such that 1 f(G) and —1 f (G). Then there exists an F E 0(G) such that f = cos F. Proof. Since 1 — f 2 has no zeros in G, there exists a function g G 0(G) ) f2 g2 = 1. Then f + ig has no zeros in G. such that (f + ig)(f —
234
10. The Theorems of Bloch, Picard, and Schottky
and hence f + ig = eiF with F E 0(G). It follows that f — ig = f 1 f,,iF e-t F ) = COS F. 2 ku
e -iF ; thus
With the aid of the lemma, we now prove: Theorem. Let G C C be simply connected, and let f E 0(G) be such that 0 f (G) and 1 f (G). Then there exists g E 0(G) such that
(1)
1 f = -2-[1 + cos r(cos rg)1.
If g E 0(G) is any function for which (1) holds, then g(G) contains no disc of radius 1. Proof. a) The function 2f — 1 omits the values ±1 in G. Thus, by the lemma, we have 2f — 1 = cos 7rF. The function F E 0(G) must omit all integer values. Hence there exists a g E 0(G) with F = cos rg. b) We set A := log(n + Vn2 - 1), M E Z. n E NVOI} and prove first that A n g(G) = 0. Let a := p±i7r log(q + Vq2 - 1) E A; then cos ira = (era + e a) = ( —1)P [(q + Vq2 - 1) -1 + (g + Vq2 - 1)1 = (-1)Pq. Hence cos r(cos 7ra) = +1 in the case p, g E Z. Since 0,1 V f (G), the set g(G) n A is empty. The points of A are the vertices of a "rectangular grid" in C. The "length" of every rectangle is 1. Since
log(n + 1 + \/(n + 1) 2 — 1) — log(n + Vn2 — 1) = log
1 + + -\/1 + n
1+ < log (1 + ;11 + 1/1 +
-
T!2-
< log(2 + id) < 7r
by the monotonicity of log x, the "height" of each rectangle is less than 1. Thus for every w E C there exists an a E A such that 'Re a — Re wl < 1/2 and lIm a — Im wl < 1/2, i.e. la — wl < I. Every disc of radius 1 therefore intersects A. But g(G) n A = 0; hence g(G) contains no disc of radius 1. 0
In the first edition of this book, Landau's equation f = — exp[ri cosh(2g)] was used instead of (1). The presentation with the iterated cosine seems easier and more natural; it was given in 1957 by Heinz Ki5nig [K], who at that time was not familiar with the second edition of [L4 ] and used Schottky's theorem in the proof.
2. Proof of Picard's little theorem. We have f = + cos7r(cos irg)) by Theorem 1, where g E 0(C) and g(C) contains no disc of radius 1. By the corollary to Bloch's theorem (see the introduction to §1), g is constant. Hence f is also constant.
§2. Picard's Little Theorem
235
Remark. Picard's little theorem can also be formulated as follows: Suppose that f, g E 0(C) and 1 = ef + e 9 . Then f and g are constant. This statement is equivalent to Picard's statement. (Prove this.) Exercises. 1) Let f,g.hE 0(C). Then the following assertions are true. a) If h = ef + e 9 . then h has Either no zeros in Cor infinitely many zeros in C. b) If h is a nonconstant polynomial, then he assumes every value. Hint for a). Transform the problem and apply Picard's little theorem to g - f.
2) Construct a function f G 0(E) that maps E locally Inholomorphically onto C. Hint. Set h(z) := zeg k(z):= 4z/(1 - z) 2 (the Koebe function), and f := h o k. -1] and h((-oc, -1)) = h((-1,0)). Show that k(E) =
Glimpse. A "three-line proof" of Picard's little theorem is possible if one knows that there exists a holomorphic covering u : E --+ {0, (uniformization). For one can then, by a general principle from topology, lift every holomorphic map f : C CVO,* to a holomorphic map f C E with f =uof. Since f is constant by Lionville, f is constant. There is also a proof of Picard's little theorem by means of the theory of Brownian motion; cf. R. Durrett: Brownian Motion and Martingales in Analysis, Wadsworth, Inc., 1984, 139-143. :
3. Two amusing applications. In general, holomorphic maps f : C C have no fixed points; f (z) := z+ez , for example, has none. But the following does hold. Fixed-point theorem. Let f : C C be holomorphie. Then fof :C always has a fixed point. unless f is a translation z z + b. b O. Proof. Suppose f o f has no fixed points. Then f also has no fixed points, and it follows that g(z) := [f (f (z)) - z] [f (z) - z] E 0(C). This function omits the values 0 and 1 (!); hence, by Picard , there exists a c E C\ {0, I } With f(f(z))- z = c(f(z)- z), z E C.
Differentiation gives (z)[fl (f (z))- =-- 1- c. Since c 1. f' has TIO zeros and f ( f (z)) is never equal to c. Thus f' o f omits the values 0 and c 0: by Picard, f' o f is therefore constant. It follows that f' = constant, hence that f(z) = az + b. Since f has no fixed points, a = 1 and b O. Exercise. Show that every entire periodic function has a fixed point.
The entire functions f := cos o h and g := sin o h, where h E 0(C), satisfy the equation f2 g2 = 1. (It is easy to show that these are all the solutions by entire functions of the equation (f + ig)(f - ig) = 1.) We investigate the solvability of the Fermat equation X' + Y" = 1. n > 3, by functions that are meromorphic in C.
236
10. The Theorems of Bloch, Picard, and Schottky
Proposition. 11 1,9 E M(C) and f n gn = 1 with n E N, n > 3, then either f and g are constant or they have common poles.
Proof Let P(f)
n
P(g) = 0. It follows from the equation f n gn = 1 that P(f) = P(y); hence 1,9 E 0(C). Suppose that g 0 O. Since Z(f) n Z(g) = 0, flg E M(C) assumes the value a E C at w E C if and only if f (w) = ag(w). It now follows from the factorization
1 = 11( f — ( v g), Çi , , (n the n roots of Xn +1, that f I g assumes none of the n distinct values (1 , Picard implies that f = cg with a constant c (en + 1)9" = 1. Hence g, and therefore f, is constant.
.
,(n . Since n > 3,
. It follows that
Remark 1. The statement just proved is representative of theorems of the following type. One considers polynomials F(zi, z2) of two complex variables, for instance — z2 — 1, and their zero sets X in C2 . If the unit disc E is the universal cover of the "projective curve" X , there exist no nonconst ant functions f, g E 0(C) such that F(f, g) = 0.
The hypothesis is satisfied if and only if the curve X has genus g > 1; the genus of the Fermat curves zr + z2 — 1 is g = 1-(n — 1)(n — 2). Remark 2. There exist nonconstant functions f,g E .A4 (C) with common poles such that f3 + g3 = 1. Indeed, the equation X 3 + Y3 = 1 describes an affine elliptic curve in C2 , the universal cover of the pro3ective curve is C, and the projection of C onto the curve determines such functions f, g. Anyone familiar with the Weierstrass p-function can give such functions explicitly. Starting with
(a + bp') 3 + (a — bp/ ) 3 =p 3 with constants a, b E C leads immediately, because of the differential equation p'2 = 4p3 — g2p — 93, to
24ab2 = 1, D = 0, 8a3 = g3. Now it is well known that the value D = 0 corresponds to the "triangular lattice" fm + ne 2 '/ 3 : m, n E Z1 (and g3 = ±r ( / (2706). Thus, if we choose the p. function corresponding to this lattice, we have f3 + g3 = 1 for
A
f :=
§3.
a + bp'
,
g :=
a — bp'
) 18
with a :=
1 2
b :=
1
Schottky's Theorem and Consequences
The growth of holomorphic functions that omit 0 and 1 can be estimated by a universal bound. Let S(r) denote the set of all functions f E 0(E) with 11(0)1 < r that do not assume the values 0 and 1. We choose any
§3. Schottky's Theorem and Consequences constant 8> 0 for which Bloch's theorem holds (0 = (0,1) x (0, oo), we consider the positive function
L(6, r) := exp [71- exp 7r (3 + 2r +
237
for instance). In
)1 0( 1 — e) ) J • 0
Schottky's theorem. For any function f E Mr),
If (z)1 < L(e,r)
for all z E E with Izi < 6, 0< e < 1.
It may seem surprising at first glance, but this peculiar theorem is stronger than Picard's little theorem, as we see in Subsection 2. In Subsection 3 we use Schottky's theorem to obtain substantially sharpened versions of Montel's and Vitali's theorems. Of course, the explicit form of the bounding function L(0, r) is not significant; our L(0, r) can be considerably improved.
1. Proof of Schottky's theorem. The proof works, with the aid of Bloch's theorem, by a clever choice of the function y in Theorem 2.1. We first observe:
(*) If cos ira = cos -n-b, then b = ±a +2n, n E Z. For every w E C there exists avEC such that cos ry = w and l vi 5 1 + iwL Proof. Since cos ira — cos 7rb = —2 sin (a + b) sin I (a — b), the first assertion is clear. To see the second, choose y = + i3 with w = cos ry such that 101 < 1. Since 10 2 = cos2 ir + sinh2 70 and sinh2 r8 > 7 2 02 , it follows that 0 + 1w1 2 /7r 2 5 1 + lwl. (v i + 32 5. We quickly obtain a sharpened version of Theorem 2.1. Theorem. If f E 0(2) omits the values 0 and 1, then there exists a function g E 0(E) with the following properties:
1) f = [1 + cos r(cos 7rg)] and Ig(0) ( .< 3 + 21f (0)1. 2) ig(z)1 < Ig(0) ( + 810(1 — 0) for all z such that Izi < 8, 0 < O < 1. E Proof. ad 1). First, the equation 2f — 1 = cos 7i" holds with By (*), there exists abEC such that cos 7rb = 2f(0) — 1 and 1bl < 1 + 12f (0)— 11 < 2+21 f (0)1. It follows from (*) that b = ±P (0) + 2k , k E Z. For F := ±F+2k E 0(1-e), we now have 2f — 1 = cos 7F with F(0) = b. Since F omits all integer values, there exists a E 0(E) with F = cos 7r4. By (*) there exists an a E C such that cos ira = b and lai < 1 + ibi < 3 + 2 1f ( 0)1. Since cos7ra = cos7r4(0), we can pass — just as for F — to a function
238
10. The Theorems of Bloch, Picard, and Schottky
g = + 2m, with g(0) = a and F = cos n-g. Property 1) holds for these functions g. ad 2). By Theorem 2.1, g(E) contains no disc of radius 1. Since d(z,DE) > I- C when I < 8, the corollary to Bloch's theorem (see the introduction to §1) implies that /3(1 - e)Igi ( z)1 < 1, i.e. Ifi(z)1 < 11:3 (1 - 0). Hence, for < 0, all z with
g(z) - g(0)
g'(C)dc;
and
< 8/3( 1 - 0 ).
This theorem immediately yields Schottky's theorem. For all tr, I cos wl < and 1 + cos WI < ci".1 . Hence 1) and 2) imply that, for all z such that
L1
(") . If (z)1 < exp[7r exp(rIg(z )1)1
The assertion follows since
expkr exp 71- (3 +
If( (J)1
lima,/ = O.
§1. Convergence on the Boundary
245
E
A power series a y e' is called a lacunary series if there exists a sequence E N such that
(*) ai = 0
when
m, < j
1.
FIGURE 11.1.
'lb prove the theorems of Subsection 1, we consider the funct ions
g(z)
— s,„(z) f (z) ,n+ (z — wi)(z — w2), fi E N. I
Every function g, is holomorphic in S (!) and the following holds. 'By definition, f can be extended holomorphically to every point of L. The reader should convince himself that this extension determines a holomorphic function in a neighborhood of L that agrees with f in E (one argues as in the proof of the existence of singular points on the boundary of the disc of convergence, cf.
1.8.1.5).
246
11. Boundary Behavior of Power Series
M. Rieses lemma. Suppose that the power series f =
E ay e (with
radius of convergence 1) has a bounded sequence of coefficients and that f is a holomorphic extension of f to S. Then the sequence g„ is bounded in S. Proof Because of the maximum principle, it suffices to show that the sequence g„laS is bounded. We verify this directly. Let A := sup la,1 < co, M = :fi s < 00. If z = rw i with 0< r 0 such that Ign's < B, n > N. Hence it follows from (1) that SL 0 and two sequences ... and no, in N such that
'Since dy-1 =
max
o< 3 k, the sequence and all its derivatives converge uniformly in E; hence
§2. Theory of Overconvergence. Gap Theorem
f is differentiable to arbitrarily high order on
rg.
For all w, z E E , w z,
oo
f (w) - f (z) w-z
bi,(wr -1 + wr-2z + • .
2+
253
z2v - 1 )
1 00
>2—
00
E b,2' = 2 — E 1
1
>0;
1
hence 1(w) f (z). Thus E is the domain of holomorphy of f by the gap theorem. The surprising thing about this example is that singularities on OIE are quite compatible with "smooth and bijective" mapping behavior of the function there. For experts in Riemann mapping theory, however, nothing sensational is going on; all we have done is explicitly give a biholomorphic map E G that can be extended to a C' diffeomorphism E —■ G. But is an infinitely differentiable closed path that is not real analytic anywhere, since 118 1E cannot be real analytic anywhere (this follows immediately from the Schwarz reflection principle, which is not discussed in this book).
ac
4. Porter's construction of overconvergent series. Let the following be chosen in some way: a polynomial q zero 0 0;
0 of degree d with q(0) = 0 that has at least one
— a lacunary series f =
ar„z 7n. with
mv+ i > dm i, and radius of con-
vergence R E (0,00). Set
g(z) := f(q(z)) = V := {z E C: lq(z)I < R},
:= d(0, av)
E (0, OC).
Proposition. The Taylor series Eby zv of g E 0(V) about 0 E V is overconvergent: It has radius of convergence r and its sequence of sections td,k (z) = Eodnik bv e converges compactly in V. The component V of V containing 0 properly contains Br (0) and is the domain of holomorphy of
91P Proof. The key (as in the overconvergence theorem) is the equation
(*
)
tdm, (Z) =
E am „ q (z)mv,
k E N,
254
11. Boundary Behavior of Power Series
which follows since tdmk (z) is a polynomial of degree < clink and q(z)mk+ 1 has only terms azi with j > ink 4.1 > dmk. The compact convergence of the sequence in V is clear from (*). The Taylor series b„zy of g G 0(V) converges in /3,-(0) C V. If its radius of convergence were greater than r = d(0. 0V), there would exist points r , V V such that kiwi converged. Then, by (*), 0„,,,q(v)mv would be convergent. But this is impossible because lq(v)i> R. Hence r is the radius of convergence of by e . We have V D B,.(0) (trivially) and V B1 . (0) (by Corollary 9.3.2, since q has distinct zeros). By the gap theorem, BR(0) is the domain of holotnorphy of f ; hence, by Theorem 5.3.2, 'ITT is the domain of holomorphy of g = f 0 q.
E
E
E
Ostrowski's example in 1 is a special case of the proposition just. proved. 5. On the history of the gap theorem. The phenomenon of the existence of power series with natural boundaries, discovered by Weierstrass and Kronecker in the 1860s, was given a natural explanation in 1892 by Hadamard. He proves the gap theorem in [Hai (p. 72 ff.); a simpler proof was given in 1921 by Szeg6 ([Sz], pp. 566 568). In 1927 J. L. Mordell, substituting polynomials u ,P(1+ te), gave a particularly elegant argument [M]. NI. B. Porter. however, had already had this beautiful idea in 1906, when he gave t he construction in 4 and, in passing, proved the gap theorem for the case rn„+ i > 2m,, by means of the substitution w(1 + w) ([Por]. PP. 191 192). Porter's work remained unnoticed until 1928; see the next subsection. Ostrovvski, in 1921, saw Hadamard's theorem as a corollary of his overconvergence theorem ([O], p. 150). In 1890 t he Swedish mathematician I. Fredholm. a student of Mittag-Leffler, pointed out an example like that in 3. For fixed a, 0 < lai < 1. he considers the power series
g(z) =EaV z
2
= 1 + az +a 2 z 1+a'z'
+ • •
cf. [Fr]. Because lirn v Via = 1. Fabry's gap theorem (see Subsection 7) shows v 2k tar < pc, for every k and that E is t he domain of holomorphy of g. Since since 2 V 2 14 < WI for small a, g also has the properties that were shown in 3 to hold for f. In a letter to Poincaré in 1891 . Mittag-Leffler called Fredholm's construction "un resultat assez remarquable"; cf. Acta Math. 15. 279 280 (1891). Fredholm's example was also discussed by Hurwitz in 1897 ([11u1. p. 478). Lacunary series, in the form of Fourier series, appear early in real analysis. Weierstrass reports in 1872 ([W], p. 71) that Rieman'', "in 1861 or perhaps even earlier," had presented the lacunary series
E sin n '3c-
•
n2
2
§2. Theory of Overconvergence. Gap Tla-,oreni
255
to his students as an example of a nowhere-differentiable function on R. "Leider ist der Be\Axis hierfiir von Riemann nicht ver6fientlicht worden und scheint sich audi nicht in semen Pa.pieren oder (lurch miindliche Oberlieferung erhalten zu haben." (Unfortunately, Riemann's proof of this was not published, nor does it seem to have survived in his papers or to have been passed down orally.) It is now known that Riemann's function is differentiable only at the points ir(2p + 1)/(2q + 1), p, q E Z. and that its derivative there is always (cf. [Cc] and [Snip.' Since Weierstrass was unable to prove Riemann's claim, he gave his famous
series
E h" cos(a n xr),
a > 3, a odd, 0 < b < a,
ab > 1 +
2 in 1872 as a simple example of a continuous nowhere-differentiable function ([W], pp. 72-74).
6. On the history of overconvergence. In 1921, Erhard Schmidt presented the paper 101, pp. 13-21, to the Prussian Academy; in this paper A. Ostrowski proves his overconvergence theorem with the aid of Hadamard's three-circle theorem. Ostrowski wrote at that time (footnote 2) on p. 14) that B. Jentzsch had discovered overconvergence in 1917 (PI, p. 255 and pp. 265-270). Ostrowski's theorem attracted attention at once. Ostrowski extended his result in several papers (cf. [0]. pp. 159-172, and the bibliography given there on p. 159); the -very elegantly constructed examples" of Jentzsch were highly regarded until 1928. But it had escaped the notice of interested mathematicians that Ni. B. Porter had already clearly described the phenomenon of overconverwlice in 1906. Porter's examples a [z(1 + z)]"?' ( [Porj, pp. 191 192), which we discussed in 4, are more natural than the "rather artificially constructed" examples of Jentzsch. Porter's series surprisingly, t his too remained hidden from the experts — were also studied in tue sanie year, 1906, by G. Faber in Munich. but Faber did not. particularly emphasize the property of overconvergence. Porter's examples were rediscovered by E. Comsat, who discussed them in the fourth edition of his Cours d 'analyse (vol. 2, p. 284). All this first became known in 1928, when Ostrowski published an addendum; cf. [O], p. 172. Ostrowski's proof of the overconvergence theorem is complicit( ed. In 1932 T. Estermann, in [E], saw that Porter's trick of exploiting the polynomials ulP(1 + w) leads to a direct proof.
E
7. Glimpses. The sharpest nonextendibility theorem that coin ains both Hadamard 's gap theorem and criterion 1.4 was already discovered in 1899 by E. Fabry (1856-1944). A series E a v zi". is called a Fabry SerleS if him m y lv = oc. We have the following deep result. 3 Riernann's example is also discussed in the article "Rieniatin's example of a continuous nondifferentiable' function." by E. Neuenschwander, in Math. Int. 1, 40-44 (1978), which is considerably supplemented by S. L. Segal in the same volume, pp. 81-82.
256
11. Boundary Behavior of Power Series
a,,zrn- is a Fabry series with radius of Fabry's gap theorem. If f convergence R. then the disc BR(0) is the domain of holomorphy of f. Proofs can be found in [1.41 (pp. 76-84) and in [D] (pp. 127-133). Fabry, incidentally, stated his theorem only for lacunary series as defined in 1.1 ([Fabry], p. 382); the formulation given here first appeared in 1906 in [Faber] (p. 581). The following exercise shows that this version is sharper. Exercise. Prove that every general lacunary series is a Fabry series. Give examples of Fabry series that are not lacunary.
P6lya noticed in 1939 that the converse of Fabry's theorem is true; he shows ([M], p. 698): Let in, be a sequence of natural numbers with mo < mi < • • • . Suppose that every series E ag, en- has its disc of convergence as domain of holomorphy. Then lim I v = oo.
Bibliography for Section 2 DINGHAS,
A.: Vorlesungen über Funktionentheorze, Grundlehren, Springer,
1961. [E]
ESTERMANN, T: On Ostrowski's gap theorem, Journ. London Math. Soc. 6, 19-20 (1932).
[Faber] FABER, G.: Über Potenzreihen mit unendlich vielen verschwindenden Koeffizienten, Sztz. Ber. K6nigl. Bayer. Akad. Wiss., Math.-Phys. Ki. 36, 581583 (1906). [Fabry]
FABRY, E.: Sur les points singuliers d'une fonction donnée par son développement en série et l'impossibilité du prolongement analytique dans les cas très généraux, Ann. Sci. gc. Norm. Sup. 13, 3. Sér., 367-399 (1896).
[Fr]
FREDHOLM. I.: OM en speciell klass of singuliira linier, dlversigt K. Vetenskaps-Akad. F6rhandl. 3, 131-134 (1890).
[Ge]
GERVER, J.: The differentiability of the Riemann function at certain multiples of r, Am. Journ. Math. 92, 33-35 (1970) and : More on the differentiability of the Riemann function at certain multiples of 7r, Am. Journ. Math. 93, 33-41 (1971).
[Ha]
HADAMARD, J.: Essai sur l'étude des fonctions données par leur développement de Taylor, Journ. Math. Pur. App!. 8, 4. Sér., 101-186 (1892), (acmes 1, 7-92.
[Hu]
HURWITZ, A.: Über die Entwicklung der allgemeinen Theorie der analytischen Funktionen in neuerer Zeit, Proc. of the 1st International Congress of Mathematicians, Zurich 1897, Leipzig 1898, 91-112; Math. Werke 1, 461480.
§3. A Theorem of Fatou-Hurwitz-P6lya
257
.TENTZSCH, R.: Fortgesetzte Untersuchungen fiber Abschnitte von Potenzreihen, Acta Math. 41, 253-270 (1918). [1,)
LANDAU, E.: Darstellung und Begriindung einiger neuerer Ergebnisse der Funktionentheorze, 2nd ed., Julius Springer, 1929; 3rd ed., with supplements by D. GAIER, 1986.
.M1
MORDELL, L. .J.: On power series with the circle of convergence as a lira, of essential singularities, Journ. London Math. Soc. 2, 146-148 (1927). OSTROWSKI, A.: Coll. Math. Pap. 5.
IP61]
P6LYA, G.: Sur les series entières lacunaires non prolongeables. C. R. Acad. Sci. Paris 208, 709-711 (1939): Coll. Pap. 1, 698 700.
[Pori
PORTER, M. B.: On the polynomial convergents of a power series, Ann. Math. 8, 189-192 (1906).
:Sin]
SMITH, A.: The differentiability of Riemann's function, Proc. Amer. Math. Soc. 34, 463-468 (1972).
1S21
SzEG6, G.: Tschebyscheffsche Polynome und nichtfortsetzbare Potenz-reihen, Math. Ann. 87, 90-111 (1922): Coll. Pap. 1, 563-586. WEIERSTRASS, K.: tTber continuirliche Functionen eines reellen Arguments. die fiir keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen, Math. Werke 2, 71-74.
§3.
A Theorem of Fatou-Hurwitz-Nlya Fiir eine beliebige Potenzreihe UBt sich der Konvergenzkreis zur natiirlichen Grenze machen, blofi lurch geeignete Ânderung der Vorzeichen der Koeffizienten. (For an arbitrary power series, the circle of convergence can be turned into the natural boundary just by a suitable change in the signs of the coefficients.) --G. P6Iya, 1916
The natural boundary of Hadamard lacunary series is their circle of convergence. This knowledge now leads to the surprising insight that such series are by no means necessary to specify uncountably many functions with discs as their domain of holomorphy. We will prove:
Theorem (Fatou-Hurwitz-P6lya). Let B be the disc of convergence of the power series f = eu av e, e, E
0, E nality of the continuum. 4
. Then the set of all functions of the form whose domain of holomorphy s B has the cardi-
4 It is known that the set of all sequences c : N {+1, -1} has the cardinality of the continuum (binary number system): hence, in any case, there exist "continuously many" functions of the form E v av z'e = .
E
258
11
Boundary Behavior of Power Series
There is something paradoxical about this lovely theorem: Although there do exist conditions on the absolute values of the coefficients that guarantee nonextendibility (Hadamard gaps, for example). there is no condition that refers only to the absolute values of the coefficients and implies extendibility. The theorem does not assert that t here exist at most countably many functions E e„a„', E = ±1,whose domain of holomorphy is not the disc B. But F. Hausdorif showed that this always does occur if lim Vki„,1 = lirn Vkl1,1 (cf. EH]. p. 103) 1. Hurwitz's proof. We may assume that B = E. Then lim VkivI == E a„,„:;"`- of f such that m, 1 > 2nr„ and there exists a subseries I. From this Hudamard lactinar series h E 0(E), we and lim am, I construct infinitely many series h„ c 0(E), n c N, such that none of the series is finite and every term a,„, zmy appears in exactly one of them. Then
VI
Ii = ho + 111 + h2 • • in E (normal convergence of power series).
We set g := f — h and assign to every sequence î, : the series :=
quho + rhhi
• • + i)
h,
{+ 1. —1 ,
N ••
v
th, ,
e 0(E)
By normal convergence. the Taylor series for every function hi about. 0 has = ± 1. Thus it suffices to show that at most countthe form EE,,a„z", ably infinitely many functions j,, do not have the unit disc E as domain of holomorphy. lf this were not true, there \ V0111(1 exist an uncountable set of sequences h such that every function L., could be extended holomorphically to a root of unity. Since the set of all roots of unity is countable, there would thus exist two distract sequences h and h' such that ff. and fp could be extended holomorphically 10 the same root of unity. Then — f = () o h () - F
h
• .
where 0,,
(S„ —
E { —2,0,2}.
would not have the unit disc as domain of holomorphy. But since not all the vanish (because (5 (5'). and since by construction all the h„ are nifimte series, the Taylor series >_2b,,z-v of h — fp E 0(E) about 0 is an Hadamard lacunary series (as a subseries of such a series). Moreover, 1 " ObvI = 1 since Inn By Theorem 1. E is the domain of holomorphy of fi — f w . Contradiction!
Historical note. P. Fatou conjectured the theorem in 1906 ([F]. p. 400) I I pc): he wrote at that and proved it for the case lim a„ = 0 and time: "Il est infiniment probable. clue cela a lieu (huts tous les cas." (It is extremely probable that this occurs in all cases.) A. Hurwitz and G. Pélya gave different proofs of the full theorem in 1916 [FM.
§3. A Theorem of Fatou-Hurwitz-P6lya
259
2. Glimpses. In 1896, E. Fabry and E. Borel already thought that almost all power series are singular at all points of their circle of convergence, hence that holornorphic extendibility at certain boundary points is the exception. Borel saw in this a problem of probabilities. In 1929, H. Steinhaus made these notions precise and proved [St]:
E
Suppose that the power series an zn has radius of convergence 1. Furtherbe a sequence of independent random numbers that are unimore, let formly distributed in the interval [0, 11. Then, with probability 1, the power sertes a„e 2 "z" has the unit disc as domain of holomorphy (in other words, the set of sequences („),,>() E 10, li N for which the series can somewhere be extended holornorphically beyond 01E has measure zero). In 1929, it was not at all clear what "probability" and "independent random numbers" meant mathematically, and Steinhaus first had to make these concepts precise He did so by constructing a product measure on the infinite-dimensional unit cube [0, l ] with the (bounded) Lebesgue measure on each factor [0,1]. A very transparent proof of Steinhaus's theorem is due to H. Boerner [Bo]; in 1938 he gave the theorem the following suggestive form: Almost all power series have their circle of convergence as natural boundary (here "almost all" means "all up to a set of measure zero" in [0, 1r). But one can also interpret the concept "almost all" topologically and ask whether a corresponding precise formulation of the notions of Fabry and Borel is possible. This idea was successfully worked out by P6lya in 1918, in [1 ] . He showed that there is a natural topology on the space of all power series with radius of convergence 1 such that the set of nowhere-extendible power series is open arid dense in this topological space and hence, in this sense, contains "almost all" the power series of the space in question. Thus the domain of holomorphy of a "general power series" is always its disc of convergence. Further results from this circle of ideas can be found in [Bi] (pp.
91-104).
Bibliography for Section 3 [13i]
BIEHERBACH, L.: Analytzsche Fortsetzung, Erg. Math. Grenzgeb. 3, Springer, 1965.
(Bot
BOERNER, H.: Über die Hhufigkeit der nicht analytisch fortsetzbaren Potenzreihen, Sitz. Ber. Bayer. Akad. Wiss. Math.-Nat. Abt. 1938, 165-174. ,
IF]
F.ATOU. P.: Séries trigonométriques et séries de Taylor, Acta Math. 30, 335-400 (1906). HAUSDORFF. F.: Zur Verteilung der fortsetzbaren Potenzreihen, Zeitschr. 4, 98-103 (1919).
[HP]
Math.
HURWITZ, A. and G. P6LYA: Zwei Beweise eines von Herrn Fatou vermuteten Satzes, Acta Math. 40, 179-183 (1917); also in HURWITZ'S Math. Werke 1, 731 734, and in PéLYA's Coll. Pap. 1, 17-21.
260
11. Boundary Behavior of Power Series P6LYA. G.: Über die Potenzreihen, deren Konvergenzkreis natiirliche Greuze ist, Acta Math. 41, 99 118 (1918); Coll. Pap. 1, 64-83. STEINHAUS, H.: Über die Wahrscheinlichkeit dafiir, daf3 der Konvergenzkreis einer Potenzreihe ihre natiirliche Greuze ist, Math. Zettschr. 31, 408416 (1930).
[PI [St]
An Extension Theorem of Szeg6
§4.
E z"1-, m > 1 fixed, and Hadamard lacunary series E have radius of convergence 1, but the corresponding holomorphic functions behave completely differently as the boundary OE is approached: although the former can be extended to rational functions with poles at the roots of unity, the latter have E as domain of holomorphy. This situation is significant for power series with only fi nitely many distinct coefficients. Geometric series
Szegii's theorem. Let f = E a,zy be a power sertes with only finitely many distinct coefficients. Then either E is the domain of holomorphy of f or f can be extended to a rational function f(z) = p(z)/(1 - z k ), where p(z) E C[z] and k E N. This beautiful theorem is proved and discussed in this section. We may assume that f is not a polynomial. Then lim 0 .1 = 1 and the series therefore has radius of convergence 1. It suffices to prove the following:
(Sz) If E is not the domain of holomorphy of f, then from some coefficient on the coefficients are periodic; that is, there exist indices A and y, < i satisfying = ap+) for all j E N. ,
Setting P :=
au zy
and Q :=
av zv, we then have
f = P + Q + QzP-A + Qz2( '''" + - = P +Q/(1 - z 1 L -A ).
z E E.
Preliminaries for the proof of (Sz) are given in Subsection 1; Runge's little theorem (12.2.1) is used there. In Subsection 2 we prove a lemma that will yield, in Subsection 3, a surprising proof of (Sz). In Subsection 4, as an application of Szeg6's theorem, we characterize the roots of unity (Kronecker's theorem).
1. Preliminaries for the proof of (Sz). Let ;o,
s
E R be prescribed numbers with 0 < < 2ir s > 1; let 6 E [0, 1) be a variable. We denote by G6 a star-shaped domain with center 0 whose boundary F(6) consists of two concentric circular arcs -n ( t) = se", (p. < t < and -r3 (t) = (1 - 6)e j1 . < t < 27r +,p, and the two line segments 72 (1) = te', 1- b < t < s, and = tezsP , 1 - b < t < s, connecting their endpoints (see Figure 11.2). We need the following: ,
§4. An Extension Theorem of Szeg6
— 72
1 /
1
1-8
/
• .... .......
/
//
/
\ 1 1
/
I
, / ,,
261
/
A
4--4 wzi
,' 0
I
1 ,,/ ..-
z2
I / G8
//
--
FIGURE 11.2. a) (Approximation theorem) There exists a 60 > 0 such that, for every ii > 0, there is a function (depending on 7.7) R(z) = co + ci 1 z + c21 z2 + •• • + 1 zq -1 + 11 zq , q E N, satisfying r (6)
71
for all 6 with 0 < 6 < ho•
E
zy b) ( Variant of Riesz's lemma) Suppose that the power series f = has bounded coefficients and that there exists a 6 > 0 such that f has a holomorphic extension f to a neighborhood of G6. Then there exists an M > 0 such that f(z)
—
<M for all n > 1.
z n-F1 r(b)
aE, there exist by 12.2.2(1') Proof. ad a). Since 1"(0)nOE is a compact set a neighborhood U of r(o) n 0E and a function Q(z) = bo + bi lz + • • • + bk _ i l z k-i +1/, z k such that ic2i u < 1 (Runge's little theorem!). Choose r> such that r(o)n Br(0) c U and fix I E N such that, for C2(z) := z -I Q(z), 1 IQ(z)I < 1
for all z E
r(o)
with izi
r.
Then la r(o) < 1. Now let V be a neighborhood of r(o) with le-2Iv < 1. For every 77 > 0 there exists an m E N such that IRI v < 77 for R := Cr. Now choose 50 > 0 so small that r (6) C V for all 6 < hood b). There exists a compact circular sector S with vertex at 0 and corners z1, z2 such that 'Y2 and -y4 lie in S (see Figure 11.2) and f is still holomorphic in S. By Riesz's lemma 1.2, the sequence
gr,(z) :=
f(z) - sn(z) zn+i
■
z - wi)(z - tv2), n E N,
262
11. Boundary Behavior of Power Series
is bounded in S. Set A := sup la,,I; then, for all points z E -y3 ,
lg. (z)I =
lan+ 1
-I- an-1- 2Z
+"I
I
lz —
lz — w21
00 •4 4A/6. o
Thus the sequence gn is bounded on I(6). Since f(x)
s_(z) zn+ 1
gn--).(z) z(z — wi)(z — w2)
—
and lz(z — w i )(z — w2)I has a minimum > 0 on r(6), this shows that the sequence [f(z) — s n _ 1 (z)j/z 71 + 1 is bounded on r(5).
2. A lemma. Let f =
E a„z' be a power series with bounded
coefficients and radius of convergence 1 that does not have E as domain of holomorphy. Then for every e > 0 there exist agEN and numbers co, c i , ,cq_i E C such that ICOart
Cian +1 + • ' • Cq -lari+q -i
an+q l
E
for all n > 1.
Proof. We choose 60 > 0 as in 1,a). Since OE is not the natural boundary of f, there exists a domain G6 of the kind described in Subsection 1 such that f has a holomorphic extension f to a neighborhood of G6 U r(6). We may assume that 6- < bo. We choose M as in Lb) and determine the function R(z) = co + ci I z + • • • + cq_ i I zq' + 1/z as in 1,a) such that R1r(6) < 27e/ML, where L denotes the Euclidean length of r(6). Then (*) R(z)
f(z)
1.
I(6)
It is now immediately clear from the equation R(z)
f. (z) — s„_1(z) z n+i c1
-0 —
'•+
cq_ 1
1\ (a
Zq -1
an+1 an+2Z + • • ')
Zq
that the number c0a.,,, + c i an+i + • • • +cq_ i an+q _ i + an+q is the residue at zero of the function on the left-hand side. This function is holomorphic in G6 \{0}. It follows (from the residue theorem, since r(5) is a simple closed path) that
con
•
4... • +cg- 'aril-T-1
1 +a n+q '----
f
R(()
f (() — 8 n-1 (C) cn+1
n > 1.
Because of (*), the standard estimate for integrals implies the assertion. D
§4. An Extension Theorem of Szegi:i
263
The lemma says that, from the qth coefficient on, no coefficient of the Taylor series for (1 + cz + • • • + co ) f (z) has absolute value greater than c. 3. Proof of (Sz). Let d1, , dk be the pairwise distinct numbers that occur as values of the coefficients a0 ,a1 ,.... Since (Sz) is trivial for k = 1 (geometric series), we may assume that k > 2. Then
d := min Id„ n#À
>0.
Since the sequence 6/0, is bounded, Lemma 2 can be applied with E := d/3. Hence there exist q E N and numbers co , c1, ,cg _i E C such that (#)
1 3d Icoan + ci an.+1 + • • • + c q - an+ q - + a+I < -
for all n E N.
We now consider all q-tuples (an , an+i , • • , an+ q -i), n E N. Since only finitely many distinct q-tuples (namely kg) can be formed from the k numbers d1 , , dk, there exist numbers A, itt E N with A < it such that (am , am-}-1, • • • ,ap+q-1)•
(a),,ax+i, • • ,c0,-+- q-i)
Since a),±3 = ap+3 for 0 < j < q, it now follows from the inequality (#) that (q-1
q-1
laX+q
==.
E cai), +3 +
-
By the choice of d, this implies that
Eci ap + , ±a,i +q CLA-F. q = CL A -F q
< ld+ld 1, then f can be extended to a rational function. 2) If p(G) = 1 and OG is a simple closed path, then either f cannot be extended holomorphically beyond OG anywhere or else f can be extended to a rational function.
From this deep theorem, which combines such heterogeneous properties as the rationality of a function, the integrality of its Taylor coefficients, and the conformal equivalence of domains, one easily obtains the P61ya-Carlson theorem by observing that p(G) < p(G') when G G' (cf. 8.4.3). A beautiful presentation of this circle of problems can be found in [ID] (pp. 192--198).
266
11. Boundary Behavior of Power Series
Bibliography for Section 4 [13]
BIEBERBACH,
L.: Analytische Fortsetzung, Erg. Math. Grenzgeb. 3, Springer,
1955.
[c
]
CARLSON. F.: Über Potenzreihen mit ganzzahligen Koeffizienten, Zeztschr. 9, 1 13 (1921).
Math,
[F]
FATOU. P.: Séries trigonométriques et séries de Taylor, Acta Math. 30. 335-40() (1906).
[H]
HAUSDORFF, F.: Zur Verteilung der fortsetzbaren Potenzreihen, Zeitschr. 4, 98-103 (1919).
[K ]
KRONECKER, L.: Zwei SAtze über Gleichungen mit ganzzahligen Coeffizienten, foam. verne angew. Math. 53, 173-175 (1857): Werke 1, 103-108.
!Pi
P61,YA, G: Coll. Pap. 1.
[PSz]
P6LYA. G. and G. SzED5: Problems and Theorems zn, Analysis, vol. 2. trans. C. E. B1LLIGHEIMER, Springer, New York, 1976.
ISz)
SZEG6, G.: Coll. Pap. 1.
Math.
12 Runge Theory for Compact Sets
Runge approximation theory charms by its wonderful balance between freedom and necessity.
In discs B. all holomorphic functions are approximated compactly by their Taylor polynomials. In particular, for every f E 0(B) and every compact
set K in B, there exists a sequence of polynomials pr, such that lim = 0. In arbitrary domains, polynomial approximation is not always possible; in C x , for example, there is no sequence of polynomials p„ that approximates the holomorphic function 1/z uniformly on a circle '7, for it would then follow that
27ri =
f lcç = lim f p„(()d( = 0. 1 7
The problem of polynomial approximation is contained in a more general approximation problem. Let K C C be a compact set. A function f : K C is called holomorphic on K if there exist an open neighborhood U of K and a function g that is holomorphic in U and satisfies !AK = f. For regions D D K, we pose the following question: When can all functions holomorphic on K be approximated uniformly by functions holomorphic in D? 'More generally: A function f that is holomorphic in a neighborhood of a circle 'y about c can be approximated uniformly by polynomials on -y if and only if there exist a disc B about c with -y c B and a function f c 0(B) such that fh, = f . The proof is left to the reader.
268
12. Runge Theory for Compact Sets
The example K = OE, D = C, shows that such approximation is not always possible: Runge theory, named after the Gottingen mathematician Carl Runge, answers the question definitively. Our starting point is the following classical theorem. Runge's approximation theorem. Every function holomorphic on K can be approximated uniformly on K by rational functions with poles outside K.
Since the location of the poles can be well controlled, this leads to a surprising answer to the question above (cf. Theorem 2.3). Every function h,olomorphic on K can be approximated uniformly by functions holomorphic in D if and only if the topological .space D\K has no connected component that is relatively compact in D.
This contains, in particular: Runge's little theorem. If C\K is connected, then every function holomorphic on K can be approximated uniformly on K by polynomials.
This approximation theorem, odd at first glance, will be obtained in Section 2 from a Cauchy integral formula for compact. sets by means of a "pole-shifting method." These techniques will be set. up in Section 1. Runge's little theorem already permits surprising applications; we use it in Section 3 to prove, among other things, — the existence of sequences of polynomials that converge pointwise to functions that are not, continuous everywhere; — the existence of a holomorphic imbedding of the unit disc in C3 .
§1.
Techniques
The Cauchy integral formula for discs is sufficient to prove almost all the fundamental theorems of local function theory. For approximation theory, however, we need a Cauchy integral formula for compact sets in arbitrary regions. Our starting point is the following formula, which was mentioned in 1.7.2.2. Cauchy integral formula for rectangles. Let R be a compact rectangle M a region D. Then, for every function f E 0(D),
1 T71-
f
jail
—
o
if z E R, if z tl R.
§1. Techniques
269
In what follows, we apply this formula to rectangles parallel to the axes. In Subsection 1 we obtain the Cauchy integral formula for compact sets, which is fundamental for Runge theory. The structure of the Cauchy kernel in the integral formula suggests that we try to approximate holomorphic functions by linear combinations of functions of the form (z - w11 ) -1 . The approximation lemma 2 describes how to proceed. In Subsection 3, we finally show how the poles of the approximating functions can be "shifted out of the way." 1. Cauchy integral formula for compact sets. Let K 0 be a compact subset of D. Then in D\K there exist finitely many distinct oriented horizontal or vertical line segments a 1 , , o- " of equal length such that, for every function f E 0(D), f (z) =
(1)
I 27ri p=1
-z
d(,
z E K.
Proof. We may assume that D C. Then o := d(K, a D) > 0. 2 We lay a lattice on the plane that is parallel to the axes, consists of compact squares, and has "mesh width" d satisfying < 6. Since K is compact, it intersects only finitely many squares of the lattice (see Figure 12.1); we denote Qk. We claim that them by (2 1 , ,
Kc
UQ cD.
The first inclusion is clear. To show that Qk c D, let us fix a point c, E QK n K. Then B6(c) C D by the definition of 6. Since the square Q has inf{la — bl : 2 The distance between two sets A. B 0 is denoted by d(A, B) a E A. b E B } . If A is compact and B is closed in C, then d(A, B) > 0 whenever
A
n B = 0.
11.111111..
11•411•1111.111•••&.
••••••11•1111•K 11•11.4•1011•EM••• • MAN- •••11.1 MIKMM • AMMO •••2111M1111 •••••••5.111IP ■
siu••or
1m- • sows. •
FIGURE 12.1.
■
270
12. Runge Theory for Compact Sets
diameter ■/.-d, the distance from c, of every point in Q" is < f2-d. Since V2d < 6, it follows that Cr c /36(c„) C D, 1 < < k. , an, that are part of the We now consider those line segments, say al, boundaries OQ' but are not common sides of two squares QP, Qg p q. We claim that
(*)
C
D \ K.
v=i
For if K intersected a segment ai, then the two squares of the lattice with ai as a side would have points in common with K, contradicting the choice of the segments a l , ... Since the common sides of different squares of the lattice occur in their boundaries with opposite orientations, it follows that k
k
E_. fa k t,p_ cl( - tr —z —
x
Q
f tf (() d( for all z E D\ U K2
,..1 av '.,
-.
z
If c is an interior point of some square, say c E C°2t, then
f (() JoQ , — cd(
27ri f (c)
and
f
f (C) = 0 for all k t 4
( C
by the integral formula for rectangles. Thus (1) has already been proved for all points of the set U . Now let c E K be a boundary point of a Q) . square Because of (*), c does not lie on any line segment a' . Hence the integrals on the right-hand side in (1) are also well defined in this case. with lim ci = c. By what has already been We choose a sequence cl E proved, equation (1) holds for all points z := cl. That it holds for z := c follows by continuity, once we observe that the vth summand on the righthand side of (1) is a continuous function at z E D\Icej. 3
(-i( 3
Remark. The theorem was probably first presented and used as a basis for Runge theory by S. Saks and A. Zygmund in their textbook [SZ], p. 155. The integral formula (1) plays a fundamental role in what follows. For the present, it is unnecessary to know that the segments a l , ... ,an automatically fit together into a simple closed polygon; see §4. 3
We could also argue directly: first, for all /,
(1((c)
Ci
(1(
f(()
(e cl( = (ci
c) f v d( . j (C - ci)(( - c) (()
1f we now choose p > 0 such that R( - ci)(( - c)I > p on Ice I for all /, then the absolute value of the difference of the integrals on the left-hand side is bounded above by • p-1 • L(crv); thus it tends to zero as 1 increases. cl
§1. Techniques
271
2. Approximation by rational functions. We begin with a lemma. Lemma. Let o- be a line segment in C disjoint from K and let h be continuous on loi. Then the function 17 h(C)(( - z) -1 c1(, z E C\Icrl, can be approximated uniformly on K by rational functions of the form in
ci, ... , cm E C, wl, — , Wm E lai.
L 'd Z — 1 1 ) 13
Proof. The function v((, z) := h(()/(( - z) is continuous on loi x K. Since Icrl x K is compact, v is uniformly continuous on lal x K; hence for every E > 0 there exists a 6 > 0 such that
for all ((,(',z) E 01 x icri x K with 1( - 0 _.< 6.
We subdivide a into segments irl,... , win of length < 6 and choose wo E lel. With co := -h(w o ) Log dC, we then have, for z E K (standard estimate),
IL
C
Z -
=
''' Wii,
Ls
E C(K), it now follows, since L(a) =
For q(z) := E 7_3. c i (z - w 1,) E 1 L(7 01 ), that
I
v(((, z) - v(w 0 , z))cl( < e • Lerth).
< L(cr) • e for all z E K.
v((, z)d( - q(z)
0
The lemma and the integral formula (1) now easily imply the basic Approximation lemma. For every compact set K in a region D, there exist finitely many line segments a1 ,... ,an in D\K such that every function f E 0(D) can be approximated uniformly on K by rational functions of the form k
II
CK
E z - w„ '
CK
K=1
E C, w„ E U Icel. v.i
Proof. We choose line segments a', ... , an in D\K , as in Theorem 1, such that 1(1) holds. By the lemma, for a given e > 0 there exist functions in,
q,,(z) = tt=1
such that
1
L
cAy
E icru l,
Z — W mv
< -,-e.1 ,
f(C) cl( - qv(z)
27ri L. ( - z
K
1
1. 8E
In general:
> O. Then Let Z be a component of C\K ,ae Z,b K u Z, and 6 = iz 1(z - a) -1 - g(z)IK > 6 -1 for every holomorphic function g that is a nonconstant polynomial in (z b ) and for which liniz - œ g(z) = O. Proof. First observe that OZ C K (cf. 2.3(1)). Now, if there were a g with < 6 , wewould have I1-(z-a)g(z)IK < 1. Since 0Z C K and lim g(z) = 0, it would follow from the maximum principle that 11- (z -a)9(z)lz < 1, which is absurd because a E Z.
§2.
Runge Theory for Compact Sets
For every compact set K C C, the set 0(K) of all functions holomorphic in K is a C-algebra. (Prove this.) We first prove that every function in 0(K) can be approximated uniformly on K by rational functions with poles outside K: the location of these poles is given more precisely in Subsection 1. We obtain as a special case that, if the space C\K is connected, every function in 0(K) can be approximated uniformly by polynomials. With the techniques of §1, the proofs are easy. In Subsection 3 we prove the main theorem of Runge theory for compact sets. 1. Runge's approximation theorem. For every set P C C. the collection Cp[z] of rational functions whose poles all lie in P is a C-algebra, and C[z] C Cp [z] C 0(C\P). The significance of the algebras Cp Izi for Runge theory lies in the following theorem. Approximation theorem (Version 1). If P C C\K intersects every bounded component of C\K , then every function zn 0(K) can be approximated 'uniformly on K by functions in Cp[z].
Proof. Let f E 0(K) and let E > O. Since f is holornorphic in an open neighborhood of K by the approximation lemma 1.2 there exist a compact
274
12. Runge Theory for Compact Sets
set L in C, disjoint from K, and a function
E Z - w, ,
q(z)
, wk E L,
with
K= 1
such that If - q1K < E/2. Let Z„ be the component of C\K containing w„. If Z, is bounded, there exists by hypothesis a t, E P n . By the pole-shifting theorem 1.3, there is then a polynomial g, in (z - t„) -1 such that
z„
(*
ctç )
Z Wk
g(z
1
)
1, Proof. By (1), there exists a polynomial P(z) = 1 + a i z + - • + akz k with 1PIK < 1. Set Q(z) := P(z)/z k ; then the inequality PIK < 1 also holds. By continuity, there exists a neighborhood U of K such that IQIu 0, M > 0, there exists a polynomial p such that
l UK + plA, < E, 1 < k < k, and minfivx(z) +p(z)i : z E B,} > M, is À < 1.
Proof. Let K := A 1 U • • U Bt. Since A1 1 , B1 are pairwise disjoint, the function h defined to be u, on AK and vA-M-C on BA is holomorphic on K. Hence, by Runge, there exists a polynomial p such that 1h + pIK < E. This means that luk +/A < E, 1 < #c < k. Moreover, since vA +p=M+E+h+p on BA, it follows for all Z E BA, 1 < À < 1, that [UAW
+p(z)1 M E — Ih(z)+ p(z)l
M.
0
One can also approximate and interpolate simultaneously. For example, let 0 be any positive divisor on C with finite support A, and let K denote a compact set with A C K such that C\K is connected. Then the following holds.
276
12. Runge Theory for Compact Sets
(3) If e > O. then for every f E 0(K) there exists a polynomial p such that
If -MK
E
and
oa (f — p) > 0(a)
for all a E A.
Proof. Choose 17) E C[z] such that oi,(f — 1 > D(a). a E A. Then, setting we have F := (f —17;)1q E 0(K). We may assume q(z) := ri„En(z that IqIK 0, since the case A = K is trivial. Hence. by Runge, there exists a is a polynomial with the desired - I . Now p := E C[z] with IF—AK < elql if properties.
3. Main theorem of Runge theory for compact sets. The approximation theorem 1 (version 2) will be strengthened. We first note two simple statements:
(1) For every component Z of D\K we have DnaZ c K. If, in addition, Z is relatively compact' in D, then Iflz < If 1K for all f e 0(D). Proof If there were acE Dn (9Z with c K, then there would exist a disc B C D\K about e. Since ZnB 0, B would then be contained in Z (since Z is a component of D\K), contradicting c E DZ. Thus D n c K. If Z is relatively compact in D, then aZ C D; hence aZ C K. The estimate now follows from the maximum principle for bounded domains.
az
(2) Every component Zo of C\K that lies in D 2S a component of D\K . If, in addition, Zo is bounded, then Zo is relatively compact in D. Proof Since Z0 is a domain in D\K, there exists a component Z1 of D\K such that Zo c Z1 . Since Z1 is a domain in C\K, it follows from the maximality of Zo that Zo = Z1. If Zo is bounded, then Zo := Zo U is compact. Since no C K (by (1), with D := C), it follows that Zo C DUK c D.
azo
We now prove: Theorem. The following statements about a compact set K in D are equivalent. i) No component of the space D\K is relatively compact in D. ii) Every bounded component of C\K intersects C\D. iii) Every function in 0(K) can be approximated uniformly on K by rational functions without poles in D. iv) Every function in 0(K) can be approximated uniformly on K by functions holomorphic in D. y) For every c E D\K there exists a functi6n h E 0(D) such that
1 11 (0 > 111 1K • 5 A subset M of D is relatively compact in D if there exists a compact set L c D with M c L.
§2. Runge Theory for Compact Sets
277
Proof. i) ii). Clear by (2). — ii) iii). This is approximation theorem iv). Trivial. — iv) i). If D\K has a component 1 (version 2). — iii) Z that is relatively compact in D, let a E Z and 6 := lz — aIK E (0, co). For (z — a ) l E 0(K), there exists a g E 0(D) such that.
1(z — or' — g(z)1K < ri;
hence
11 — (z — a)g(z)1K < 1.
By (1), it follows that 11 — (z — a)g(z)1 < 1 for all z E Z, which is absurd for z = a. i) v). The region D\(K U {c}) has the same components as D\K , except one from which c has been removed. Hence 1) holds and thus, by what has already been proved, iv) also holds for K U {c} instead of K. Thus, corresponding to the function g E 0(K U {c}) defined by
g(z) := 0
for z G K, g(c) := 1,
there is an h E 0(D) such that 1hIK < and 11 — h(c)1 < 1. This implies that Ih(c)1 > t- and proves v). v) i). If D\K had a component Z that was relatively compact in D, then, by (1), v) would fail for every point c E Z. For D = C, we obtain a more precise form of Runge's little theorem. Corollary 1. Every function in 0(K) can be approximated uniformly on K by polynomials if and only if C\K is connected. This occurs if and only if for every c E C\K them exists a polynomial p such that 1p(c)1 > 1P1K • Proof. C\K is connected if and only if C\K has no component that is relatively compact in C. Thus the corollary follows immediately from the theorem, since every entire function in C can be approximated compactly by Taylor polynomials.
We now also have the converse of the first version of the approximat ion theorem 1. Corollary 2. If P C C\K is such that every function in 0(K) can be approximated uniformly on K by functions in Cp [z], then P intersects every bounded component of C\K Proof. By the pole-shifting theorem 1.3, we may assiune that P intersects every component of C\K in at most one point. Then D := C\P is a region with K c D. Since Cp [Z] C 0(D), every bounded component of C\K inii). tersects C\D = P because of the implication iv) With the aid of (2). the equivalence i) 0} and prove: Proposition. There exists a sequence of polynomials p n with the following properties: 1)
2)
lim p(0) = 1, limp(z) = 0 for every point z E
.
lim p(z) = 0 for every point z E C and every k > 1. 1L OC
p yc )
p 2( k )
, k E N, converges compactly in C\R± , but none of these sequ,ences converges compactly in any neighborhood of any point in
3) Every sequence
Proof We set,
in :=
{Z
E T3,,(0) and d(z,
)
.717.1 , Kn
{0} U [ -7, 711 , n] UI, n > 1.
§3. Applications of Runge's Little Theorem
279
Kn is compact and every set C\ KT" is connected (see Figure 12.2). We place , n], respectively, in such a compact rectangles R, and Sn around 0 and way that the following holds.
FIGURE 12.2.
The sets R n , Sn , and In+1 are pairivise disjoint. The compact set L n := U in+i is a neighborhood of K„; that is, K„ c L n . The set C\L, is connected.
The function gn defined by
g,1 (z) := 0 for z E Ln\Rn,
g„(z) := 1 for z E
is holomorphic in L n . Hence, by Runge, there exists a polynomial pn such that 1 Pu
(*)
Since gi„ satisfies
(**)
gniL,
0, then the sequence would be uniformly convergent on 0.131 x 1(0). By the maximum principle,
ye
This follows immediately from the Cauchy estimates for derivatives in compact sets; cf. 1.8.3.1. 6
280
12. Runge Theory for Compact Sets
it would also be compactly convergent in B 1.,;1 (0), which is impossible. It. now follows that none of the sequences of derivatives converges compactly about points of R+ ; cf. 1.8.4.4.
Historical note. In 1885, Runge constructed a sequence of polynomials that converges pointwise everywhere in C to 0, without the convergence being compact in all of C. He showed, "an einem Beispiel einer Summe von ganzen rationalen Functionen , dass die gleichtnAssige Convergenz eines Ausdrucks nicht nothwendig ist, sondern dass dieselbe auf irgend welchen Linien in der Eberle der complexen Zahlen aufhiiren kann, wahrend der Ausdruck dennoch fiberall convergirt und eine monogene analytische [ = holomorphe] Function darstellt" (by an example of a sum of rational entire functions , that the uniform convergence of an expression is not necessary, but rather that it can cease on any line in the plane of complex numbers, while the expression nevertheless converges everywhere and represents a monogenic analytic [ = holomorphic] function) ([Run], p. 245). Runge approximates the functions 1/[n(nz — 1)] by polynomials: his sequence "convergirt ungleichrniissig auf dem positiven Theil der imagindren Achse" (converges nonuniformly on the positive part of the imaginary axis). Runge was the first to ask whether, in the Weierstrass convergence theorem, "die gleichrrassige Convergenz nothwendig ist, damit eine monogene analytische Function dargestellt werde" (uniform convergence is necessary for a monogenic analytic function to be represented). His counterexample was hardly noticed at the time. I have yet to learn who gave the first example of a sequence of holornorphic functions that converges pointwise and has a discontinuous limit function. This was already mathematical folklore by around 1901; cf., for example, [O], p. 32. Since 1904 at the latest, it has been easy to give such sequences explicitly. It was then that Mittag-Leffler constructed an entire function F with the following paradoxical property (cf. [ML], théorème E. p. 263): (1)
F(0) 0,
lim F(re') = 0
r ---•
for every (fixed)
E [0,27).
The sequence gn (z) := F(nz) converges pointwise in C to a discontinuous limit function. — The Mittag- Leffler function F can be written down explicitly as a "closed analytic expression"; see, for example, [PS] (vol. 1, III, Problem 158, and vol. 2, IV, Problem 184). Montel, in 1907, systematically investigated the properties of such limit functions, which he called "fonctions de première classe" (Nob p. 315 and p. 326). F. Hartogs and A. Rosenthal, in 1928, obtained very detailed results [HR.]. A particularly simple construction of an entire function satisfying (1) was given in 1976 by D. J. Newman [NI. He considers the nonconst ant entire function
G(z) and proves:
g(z + 47),
where
g(z):= f e zt t -t dt, z E C,
§3. Applications of Runge's Little Theorem
281
The function G is bounded on every real line through 0 E C. Then, if G — G(0) vanishes to order k at 0, it is obvious that F(z) := [G(z) — G(0)1/z k is an entire function for which (1) holds. Remark. "Explicit" sequences in 0(C) can also be given which vanish pointwise everywhere in C but do not converge compactly to O. Let F denote the "MittagLeffler function" given by (1).* The sequence fn (z) := F(riz)In E 0(C) converges pointwise to 0 in C, but this convergence is not uniform an any neighborhood of the origin. Proof By (1), the sequence f,, converges pointwise in C to O. If it were 'uniformly convergent in a neighborhood of 0, then, in particular, it would be bounded in a neighborhood of O. There would then exist r > 0 and M > 0 such that IF(z)I < nM for all n > 1 and all z E C with Izi < 711' . For the Taylor coefficients ao , of F about 0, it would follow that lag,' < (nM)/(nr)'
for all v > 0 and all n > 1.
Since lim„—(nM)1(nr) = 0 for v > 2, F would be at most linear and hence, since F(re) = 0, identically zero, contradicting F(0) O. Exercise. Let f E 0(C). Construct a sequence of polynomials pn that converges pointwise to the function
g(z) :=
{ O 1(z)
for z E C\R for z E R.
Do this in such a way that the convergence in C\R is compact and all the sequences of derivatives p,,,k) converge pointwise everywhere in C.
2. Holomorphic imbedding of the unit disc in C3 . If , fH G the map D is called 0(D), 1 < n < ac, z 1-4 (fi(z),.. , f n (z)), 0) holomorphic. Such a map is called smooth if (f;(z),... , f",(z)) 0 (0, for all z E D. Holomorphic maps that are injective, closed, 7 and smooth are called holomorphze zmbeddings of D into C. It can be shown that the image set of D in C" is then a smooth (with no singularities) closed complex curve in Cn ( = complex submanifold of C" of real dimension 2). — Regions D 0 C admit no holomorphic imbeddings in C, since every holomorphic imbedding D C is biholomorphic. Our goal is to prove the following theorem. Imbedding theorem. There exists a holomorphic imbedding E -+ C3 . To prove this, it suffices to construct two functions f, g E 0(E) such that f (z„)1 + g(z„)1) = oc for every sequence z n E E converging to a point in 8E\111. Then the three functions f, g. and h, where h(z) := 1/(z - 1), 7A
map X
Y between topological spaces is called closed if every closed set
in X has a closed image in Y; see also 9.4*.1.
282
12. Runge Theory for Compact Sets
give a closed holomorphic map E --+ C3 . Since h : E —, C is injective and h' (z) 0 for all z, this map is also injective and smooth. The functions f and g are constructed with the aid of Runge's little theorem. We choose a "horseshoe sequence" KT, of compact pairwise disjoint sets in E such that C\ (K0 U..-UK,) is always connected and the sets Kn tend to the boundary 51E and taper toward 1 (as in Figure 12.3).
FIGURE 12.3.
Lemma. There exists a holomorphic function f E 0(E) such that min.(' f (z)1: z E KO > 2" for all n E N. Proof We choose an increasing sequence 170 C V1 C • • • of compact discs about 0 such that U V, . E and
Ko u K i u • • - u Kn-i c Vn,
Vn
n Kn — 0, n
E N.
Then C\ (V„ U K„) is connected. We now recursively construct a sequence (pa ) of polynomials. We set po := 3. Let n > 1 and assume that po , pi , ..., pn _ 1 have already been constructed. By 2.2(2) (with A1 := V„, Bi := K. u 1 := 0, v1 := po + - • • +pn—i), there exists a polynomial p„ such that (*) 1Pre I V,
2 —n ,
Mill{ 1P0(Z) ± • ' • +p, 1 (z)+p(z)1 : z E K„}
2' +1.
The series Ep, converges normally in E to a function f E 0(E), since
IA, I vr,
iPi, 1 v„ n
and therefore 00
E 1 Pv1v, < oo 1)=0
for all n > 1.
§4. Discussion of the Cauchy Integral Formula for Compact Sets
283
Since K„ C V,, for I, > n, we have 1/9,11(7, < 2' for v > n by (*). It follows from the second inequality in (*) that, for all z E K„,
f (z)I
Ipo(z) + • • • + pn_ 1(z) + pn(z)I — v>n
> 2n + 1
E 2-- > v>r
Similarly, we choose a second horseshoe sequence L„ in E such that L„ covers the ring between K„ and K71 +1 except for a trapezoid with the real axis as midline (1,7, is drawn with a dashed line in the figure). Then, by the lemma, there exists a g E 0(E) such that rnin{ Ig(z)1 : z E L„} > 2'' for all n. By the construction of f and g, it now follows immediately that, for every sequence z„ E E with lim z„ E OE\ 01,
(**)
limaf(z,)1 + Ig(z.)1)
= 90.
This completes the proof of the imbedding theorem. Remark. The construction of the functions f and g can be refined so that (**) also holds for all sequences z, E E with lim z„ = 1. H. Cartan noticed this as early as 1931 ([C], p. 301); cf. also [Rub], pp. 187-190. Thus there exist finite holomorphic maps E C2 (for the concept of a finite map, see 9.3). It can be shown that there even exist holomorphic imbeddings of E into C2 ; see, for example, [Gau ] . It can also be proved that every domain in C can be imbedded in C3 .
§4.
Discussion of the Cauchy Integral Formula for Compact Sets It is aesthetically unsatisfying that in the Cauchy integral formula, which underlies Runge theory, integrals are taken over line segments rather than closed paths. In this section we give Theorem 1.1 a more attractive form. It turns out that, although we cannot get by with a single closed path in general, there always exist finitely many such paths with nice properties. — To formulate the theorem conveniently, we introduce some terminology that is also used in the next chapter. Every (formal) linear combination -y = a1-y 1 + •
an-tn , a,, E Z,
is called a cycle in D. The support over cycles are defined by
fd( :=
at,
-y" a closed path in D,
1 < u < n,
H = 12'1 of -y is compact.
f
, f E
C(H)•
Integrals
284
12. Runge Theory for Compact Sets
The index function 1 f c1( zE Z, z E := -27ri is locally constant. The interior and exterior of 'y are defined by Int^f := {z E C\hd : ind.y (z)
0 } , E)ct'y := 1z
E
C\H : ind-y (z) = 0);
these sets are open in C, and the exterior is never empty. A closed polygon T = [pip2 • • PkPi] composed of k segments [p1,P2], [P2, p3], . . , [pk,pil is called a step polygon if every segment is horizontal or vertical in C and all the "vertices" pi, p2, pk are pairwise distinct. Then every point in T except p i is traversed exactly once.
1. Final form of Theorem 1.1. We use the notation of 1.1. Let a-1 , a" E D\K be oriented segments of a square lattice, parallel to the axes in C, for which Theorem 1.1 holds. We may assume that o-i` ±a`' for p v. The integral formula 1.1(1), applied to f(z)(z — c) with c E K, gives the integral theorem T1
E fcr" f (()d( 0
(1)
for all f E 0(D).
One consequence of this formula is that the segments c together into a step polygon. We claim:
automatically fit
Cauchy integral formula for compact sets (final form). For every compact set K 0 in a region D, there exist finitely many step polygons T 1 , T m in D\K such that, for the cycle -y := T 1 + • • • + Tm ,
(2)
f(z) =-72 1i
(
f. ((;) z
A.
for all f E 0(D) and all Z E K.
In particular, (3)
K c Int
C D and
ind,y (z) = 1
for all z E K.
Proof. We write a" = [a y , b1,[ and first prove:
(*) Every point c E C is the initial point a, exactly as many times as it is the terminal point bt, of one of the segments ,a". Let /and f denote the multiplicity with which c occurs in the n-tuples b„), respectively. We choose a polynomial p such
(a 1 ,... ,a„) and (b 1 , that
p(c) = 1, p(au ) = 0 for ("1,
c,
p(bp ) = 0 for bi,
c.
§4. Discussion of the Cauchy Integral Formula for Compact Sets
p(a,) = land Eni„. 1 p(bm ) = L. By (1),
Then > O
285
=
E
Crv
(C)d( =E[p(b„) — p(a,,)]; hence 1= L, i.e. (*).
The next statement now follows easily.
( ) There exist an enumeration of the a' and natural numbers 0 = ko < a lc o +1 is a step k 1 < • • • < km+1 = n such that rA polygon, 1 < 1.1 < m. Because of (*), finitely many closed polygons can be formed from the av . Let 6 = [61142 dkdd be such a polygon. We show by induction on k that 6 decomposes into step polygons. The case k = 4 is clear. If k > 4 and 6 is not a step polygon, then there exist indices s < t with ds = dt . Then d2 ds dt+ i (it] are closed polygons with fewer dkdi] and [dsd8+1 than k segments to which the induction hypothesis can be applied. Hence ( ) holds. By Theorem 1.1, this proves the existence of a cycle for which (2) holds. The integral theorem for -y now follows from (2), and 1/(z — c) E 0(D) for all c 0 D; hence Int -y c D. Setting f 1 in (2) gives ind.),(K) = 1. Warning. In general, there is no closed path -y in D\K with K C Int -y C D and a fortiori no closed path for which formula (2) holds. If K is a circle about 0 in D := Ex , for instance, then every closed path -y with K C Int -y lies in the exterior of K. The disc bounded by K then lies in Int so that Int -y Ex . In this example, the theorem holds for all cycles + '72 , where -y' and -y2 are oppositely oriented circles about 0 in Ex, one of which lies outside and one inside
K. Remark. That the segments of . , an in Theorem 1.1 automatically fit together into closed polygons was observed in 1979 by R. B. Burckel; in [13], pp. 259-260, he derives (*) somewhat differently. ,
2. Circuit theorem. It is intuitively clear that one can go around any connected compact set K in D by following a closed path in D\K. This statement is made precise in what follows; in addition to Theorem 1, we need the Jordan curve theorem for step polygons: Lemma. Every step polygon divides C into exactly two domains:
C\17- 1 = Int T U Ext T
and ind,(Int 7- ) = ±1.
We carry out the proof in three steps. Let T = [T) Lr1P2 • • P011) with successive segments 7„ = [p,p +. 1 ], 1 < t < k, where pk +i := pi . Around each segment irs, we place the open rectangle R, of length d ( = mesh width of the lattice) and width d/4, with 7,,as midline. In one subrectangle of
286
12. Runge Theory for Compact Sets
R 1 \ kr i we choose a point p, and in the other a point q. We denote by U and V the components of C\IT1 with pE U,qE V, and first. prove that Z PK
P2
• (I RK
7rK
TrK -1
R,
R,
PK
ITK
R„
V
FIGURE 12.4.
a) Md.,- (U) = ind,-(V) ± 1; hence, in particular, U
n v = 0.
{±ao
If Q is the square that has in as one side and contains q, let -y G be chosen in such a way that —7r 1 is a segment of -y (first part of Figure 12.4). With the auxiliary polygon 6 := [p 1 utp2 • • • pp ] , we then have k
inci h (z)
inc1 7.(z) + ind,,(z)
for all z E CV161IJ [Pi.P2])•
Since the index function is locally constant and [p, C C\1151, it follows that indl,(p) = ind6 (q). Since ind-f (p) = 0 and ind.,/ (q) = ±1, this yields We next observe that (p) = ind r (q) + 1 and therefore a).
b) R,\17 CUUVf or all k = 1, . . . ,k . Every point in R, \17r,1 can be joined to a point. in R,_ 1 \17,_ i I by a path in C\ITI, 2 < < k (the two possible situations are illustrated in the second and third parts of Figure 12.4; since T is a step polygon, neither that intersect at A, of the segments of the grid distinct from rr„_ i and belongs to T). Thus b) follows by induction since Ri \17rd c UuV trivially, by the choice of U and V. After these preliminaries, it is easy to complete the proof of the lemma. Let w E C\ITI be arbitrary. We pass a line g through w such that g 0 a,n(l g intersects no vertex Pk of T. On g, there is a point w' E r that is closest to w. Since w' is not a vertex, some point of the rectangle R, lies on the segment [w, 0]. Since 1w, w') C C\171, it follows from h) that C\ITI C U U V. The domains U and V are thus the only components of C\ITI. Since UnV = 0 by a), it follows that either U = mt T and V = Ext or vice versa. In both cases, a) implies that indr (Int T) = ±1.
n171
Remark. The proof can be modified so as to hold for all simply closed polygons; it is also easy to see that the polygon is the common boundary of its interior and exterior. — The proof above is probably due to A. Pringsheim ([Pr], pp. 41-43).
The theorem now follows from Theorem 1 and the lemma.
§4. Discussion of the Cauchy Integral Formula for Compact Sets
287
Circuit theorem. If K is a compact set in D, then for every connected subset L of K there exists a closed path (step polygon) r zn D\K such that incl,(L) = 1.
Proof. Choose -y = 7 1 + • - + -rm as in Theorem 1. Let c E L. Since 0. By 1 = ind-y (c) = indr .(c), there exists a k such that ind T h (c) the lemma, indr (c) = 1 for T := r k or T := — T k . That incl.,. (L) = 1 follows because L is connected. Remark. One cannot always find T such that Int T C D; for example, no such r exists in the case L = K := aE and D := C x .
Bibliography [131
BURCKEL.
R.
B.: An Introduction to Classical Complex Analysis, vol. 1,
BirkhAuser, 1979.
[CI
CARTAN, H.: Œuvres I, Springer, 1979.
Pail]
GAIER, D.: Vorlesungen über Approximation im Komplexen, Birkhiiuser, 1980.
iGai21
GA1ER, D.: Approximation im Komplexen, Jber. DMV86, 151-159 (1984).
(Gau ]
GAUTHIER, P.: Un plongement du disque unité, Sém. F. Norguet, Lect. Notes 482, 333-336 (1975).
[HR[
HARTOGS, F. and A. ROSENTHAL: Über Folgen analytischer Funktionen, Math. Ann. 100, 212-263 (1928).
[ML[
MITTAG-LEFFLER, G.:
[Mo]
MONTEL, P.:
[NI
NEWMAN, D. J.: An entire function bounded in every direction, Amer. Math. Monthly 83, 192-193 (1976).
Sur une classe de fonctions entières, Proc. 3rd Int. Congr. Math., Heidelberg 1904, 258-264, Teubner, 1905.
Sur les suites infinies de fonctions, Ann. Scz. Ec. Norm. Sup. (3)24, 233-334 (1907).
OSGOOD, W. F.: Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals, Ann. Math. (2)3, 25-34 (1901-1902). 1311
PRINGSHEIM, A.: Über eine charakteristische Eigenschaft sogenannter Treppenpolygone und deren Anwendung auf einen Fundamentalsatz der Funktionentheorie, Sitz. Ber. Math.-Phys. Ki. K6nigl. Bayrische Akad. Wiss. 1915, 27-57.
;PS]
P6LYA, G. and G. SzEG6: Problems and Theorems in Analysis, 2 volumes, trans. C. E. BILLIGHEIMER, Springer, New York, 1976.
,Rubl
RUBEL, L. A.: How to use Runge's theorem, L'Enseign. Math. (2)22, 185190 (1976) and Errata ibid. (2)23, 149 (1977).
288 [Run]
12. Runge Theory for Compact Sets RUNGE,
C.: Zur Theorie der analytischen Functionen, Acta Math. 6, 245-
248 (1885). [Szi
SAKS, S. and A. ZYGN1UND: Analytic Functions, trans. E. J. SCOTT, 3rd ed., Elsevier, 1971.
13 Runge Theory for Regions
Jede eindeutige analytische Function kann durch eine einzige unendliche Summe von rationalen Ftinctionen in ihrem ganzen Giiltigkeitsbereich dargestellt werden. (Every single-valued analytic function can be represented by a single infinite sum of rational functions in its whole region of validity.)
C. Runge, 1884 In Chapter 12 we proved approximation theorems for compact sets; we now prove their analogues for regions. We pose the following question: When are regions D, D' with D c D' a Runge pair? That is, when can every function holomorphic in D be approximated compactly by functions holomorphic in D'?
The pair IF?, C is not Runge since 1/2 E 0(1E 1. Since T is locally finite in D, every set Tr, := T n \K,i), n > 1, is finite. We may assume that Tn 0. Let k,, be the number of points in T. Since, for every point 4 E Tr„ the principal part q, is holomorphic in Kn , there exists by 12.2.1 a g, E 0(D) such that lq, —gIK< 2 - n /k n . Now h := E 0,0 (q, - g,) is a Mittag-Leffler series for the distribution (d, , q,): Every compact set in D lies in almost all the K n and, after omission of finitely many terms, the series is bounded above on K by E 2 - v; thus it converges normally in D\T Historical note. This proof was given in 1885 by C. Runge ([R ] , pp. 243-
244). 2. Runge's approximation theorem. As an analogue of the approximation theorem 12.2.1 1 we now have Runge's theorem on rational approximation. Let P c C\D be a set whose closure P intersects every hole of D. Then the algebra Cp [z] C 0(D) is dense in 0(D). Proof. Let K be a compact subset of D. We choose K1 as in 1(1). Then every bounded component of C\KI intersects P and hence also P. By the approximation theorem 12.2.1, every function in 0(D) c 0(K 1 ) can therefore be approximated uniformly on K1 and a fortiori on K by functions in Cp Ezi. The claim follows from the observation made in the introduction. 0
Setting P := 0 gives an immediate corollary. Runge's polynomial approximation theorem. If D has no holes, then the polynomial algebra C[z1 is dense in the algebra 0(D).
We also obtain (even if D has uncountably many holes): For every region D in C, there exists a finite or countable set P of boundary points of D such that the algebra Cp[zi is dense in the algebra 0(D). Proof. Every hole of D intersects 3 .13 (see A.1(3)). Since ap is a subspace of C, there exists a countable set P c OD with P = ar) (second countability of the topology on aD). 3. Main theorem of Cauchy function theory. In 1.9.5 it was proved, by an argument due to Dixon, that the Cauchy integral theorem holds for closed paths in D whose interiors lie in D. We alluded then to a short proof using Runge theory. We now give this proof for arbitrary cycles. The following convergence theorem holds for cycles -y just as it does for paths:
§1. R.unge's Theorem for Regions
293
(1) If a sequence f„ E C(1-y1) converges uniformly on yj , then lim f f„d( = f (lim f„)d(. A cycle -y in D is called null homologous in D if its interior lies in D: C D. We can now state and prove the
Int -y
Main theorem of Cauchy function theory. about a cycle -y in D are equivalent.
The following statements
it) The integral theorem f,), f d( = 0 holds for all f E 0(D). ii) The integral formula
ind-y (z)f (z) =
1 (f (()z d(,
zE
D\I-yi,
holds for all f E 0(D). iii) -y is null homologous in D. Proof. The equivalence i) ii) follows verbatim as for paths. The implication i) iii) is trivial, since 1/(( — w) E 0(D) for all w G C\D. The implication iii) i) is the core of the theorem and is proved as follows: Let f E 0(D) be given. By Runge's theorem 2, there exists a sequence q„, all of whose terms are rational functions with poles only in C\D, which converges uniformly to f on the support H of -y. By (1),
(*
f d( =
)
f q n d(.
Now, if Pn denotes the finite pole set of (hi , we have
1 27ri
f q„d( =
E
ind,y (oresc q,
(residue theorem).
CEPanint ^y
But Pn C C\D and Int -y C D; hence Pn n on the right-hand side of (*) vanish.
= 0 and all the integrals
4. On the theory of holes. Runge's theorems in Subsection 2 lead us to consider the holes of regions. The proof of our first result is quite easy.
(1) If is a cycle in D that is not null homologous, then the interior of -y contains at least one hole of D. Proof Since Int-y D, there exists a point a E C\D with ind.),(a) O. Since the index function is locally constant and the component L of C\D containing a is connected, ind,y (z) = ind-y (a) for all z E L. Hence L C Int Since the function ind-y vanishes when z is large, L is bounded and therefore a hole of D.
294
13. Runge Theory for Regions
Our next result follows from the contrapositive of (I). (2) If D has no holes, then D is (homologically) simply connected. The converse of (2) is true but not as easy to prove. We obtain it in 2.4 by a function-theoretic detour. In 14.1.3, we will reverse (1) and show (as is immediately evident) that for every hole L of D there exist closed paths -y with L C Int -y. The construction of such paths is difficult, since the family of holes of D can be quite unpleasant (Cantor sets). The theory of holes is complicated. The definition of holes is based on "how D lies in C"; it is not a definition by "intrinsic properties of D," and is not a priori an invariant of D. Perhaps domains with two holes could be topologically or even biholomorphically mapped onto domains with three holes. We will see in 14.2.2 that this cannot happen. Although simply connected domains in R2 C), by the discussion above, are precisely those domains that have no holes, domains in r n > 3, may have holes and still be simply connected: For instance, if (in)finitely many pairwise disjoint compact balls are removed from R", n > 3, what remains is a simply connected domain in R" with (in)finitely many holes. ,
5. On the history of Runge theory. The year 1885 marks the beginning of complex and real approximation theory: Runge published his groundbreaking theorem — our Theorem 2 — in Acta Mathematica [R]; Weierstrass, in the Sitzungsberichte der Kaniglichen Akademie der Wissenschaften zu Berlin, published his theorem on the approximation of continuous functions by polynomials on real intervals [IN]. Runge's approximation theorem did not receive much attention at first; only since the 1920s has it decisively influenced the development of function theory (cf., for example, [C]). Runge approximates the Cauchy integral by Riemann sums. In doing so, he already obtains rational functions, which, however, still "become infinite at points where the behavior of the function is regular. To remedy this situation," he develops the pole-shifting technique, whereby he moves the poles to the boundary. Runge says nothing about approximation by polynomials, although his method also contains the theorem on polynomial approximation. No less a mathematician than D. Hilbert proved this important special case by different means in 1897 [Hi]; Hilbert does not mention Runge's work. The motive for Runge's investigations was the question whether every domain in C is a domain of holomorphy. With his approximation theorem, he could give an affirmative answer (see also .5.2.5); he showed moreover that his approximation theorem subsumed that of Mittag-Leffier, proved two years earlier. Even today, a hundred years later, Runge's method provides the easiest approach to complex approximation theory. Other approaches to Runge theory can be found in [FL], [Ho], and [N].
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295
Runge's theorem was generalized in 1943 by H. Behnke and K. Stein. In their paper [BS], which because of the war did not appear until 1948, they consider arbitrary noncornpact Riemann surfaces X instead of C; they prove (cf. p. 456 and p. 460):
For every region D in X, there exists a set T of boundary points of D (in X) that is at most countablé and has the following property: Every function in 0(D) can be approximated compactly in D by functions rneromorphic in X that each have only finitely many poles, all of which lie in T. This theorem can be used to show, for example, that noncompact Riemann surfaces are Stein manifolds.
§2.
Runge Pairs
In this section, D, D' always denote regions in C with D c D', We call D, D' a Runge pair if every function in 0(D) can be approximated compactly in D by functions h1D, h E 0(D') (the subalgebra of 0(D) resulting from the restriction 0(D') 0(D), h hID, is then dense in 0(D)). In Subsection 1, we characterize such pairs topologically by properties of the (unpleasant) locally compact difference D'\D; its components play a crucial role. We need Sura-Bura's theorem from set-theoretic topology (see the appendix to this chapter). ln Subsection 2 we consider Runge hulls; in Subsection 3, following Behnke and Stein, we characterize Runge pairs by a homological property. In Subsection 4, those functions that can be approximated will be described by an extension property. Runge pairs D, D' with D' = C are of particular interest; D is then called a Runge region. In Subsection 5 we prove, among other things, that every Runge domain C can be mapped biholomorphically onto the unit disc.
1. Topological characterization of Runge pairs. We begin by proving two lemmas. (1) Every component L of C\D with L c D' is a component of D'\D. Proof. We have L c D'\D. Since L is connected, there exists a component L' of D'\D with L' D L. Since L' c C\D arid L' is connected, it follows from the definition of components that L' = L.
(2) For every open compact subset A of D'\D, there exists a set V that is open and relatively compact in D' and satisfies A c V and aV c D. Proof. We have D'\D = AU B, AnB = 0, where B is closed in D'\D. Since D'\D is closed in D', B is also closed in D'. Hence there exists a covering of A by discs that are relatively compact in D'\B. By Heine-Borel there
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13. Runge Theory for Regions
exists a set V that is relatively compact and open in D' and satisfies A C V,
VnB=0. It follows from av n B = O But av C D'; hence av C D.
= av n A that av n (Df\D) = 0.
The next theorem is an easy consequence of (1), (2), and Corollary A.2.1. Theorem. The following statements about regions D, D' with D C D' are equivalent:
0 The space D'\D has no compact component. ii) The algebra of all rational functions without poles in D' is dense in 0(D). iii) D, D' is a Runge pair. iv) In the space D'\D, there is no open compact set O. Proof. 0 = ii). By (1), the set P := C\D' intersects every hole of D. Since P C C\D, Theorem 1.2 implies that the algebra Cp [Z] is dense in 0(D). ii) = iii). This is clear, since rational functions without poles in D' are in 0(D'). iii) iv). Let A be an open compact subset of D'\D. We choose V as in (2); then aV is compact and av C D. Suppose there were a point a E A. Then (z — a ) l E 0(D); choose a sequence gn E 0(D') with lim 1(z — — gn iov = O. But then lim — (z — a)g n (z)lew = O. Since V C D', the sequence (z — a)gn (z) would converge uniformly to 1 by the maximum principle, which is impossible because aE A c V. Hence A is empty. iv) = i). This is clear by Corollary A.2.1.
Behind the purely topological implication iv) = i) lies Sura-Bura's theorem; see the appendix. In general, compact components of D'\D are not open in D'\D, as is shown by the examples D := D'\Cantor sets. D We now also have the converse of Theorem 1.2. Corollary. If P C C\D is such that the algebra Cp [z] is dense in 0(D), thenTP intersects every hole of D. Proof. D C C\P and Cp [X] C 0(CVP); hence D, C\P form a Runge pair. It follows from the implication iii) = i) that (C\TD)\D = CVP U D) has no compact component. But this means that P intersects every hole of D. 0
2. Runge hulls. Let D' be a given fixed region in C (total space). For every subregion D of D', we set
:= DU RD,
where RD
:=
union of all open compact subsets of Di\D.
The set RD can be quite pathological, e.g. a Cantor set; see A.1.4). By
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297
Corollary 1 of A.2, RD is the union of all the compact components of D'\D. The next statement now follows from A.3(1). (1) The set b is a subregion of D'. The difference D'\ b - contains no open compact set. is called the Runge hull of D (in D'). When D' = C, we obtain .5 from D by "plugging up all the holes of D." We now justify the choice of the term "Runge hull." Proposition. The pair D, D' is Runge. 13 = D if and only if D, D' is a Runge pair. For every Runge pair E, D' with D C E, we have D c E. Proof. The first two statements are clear by (1) and Theorem 1. — If Dc Ec D', then D'\E is closed in D'\D. Thus (D'\E) n K is compact and open in D'\E for every open compact subset K of D'\D. Since E, D' is a Runge pair, Theorem 1 implies that (D'\E) n K is empty. It follows that K c E; therefore RD=UKCEalldb=DURDC E.
We see that f) is the "smallest region between D and D' that is relatively Runge in D'." It follows immediately from the proposition that
r")
=D ,
DcE(cD')=bc E.
3. Homological characterization of Runge hulls. The Behnke-Stein theorem. Let the regions D, D' be given, with D C D'. We denote by .7 the family of all cycles 'y in D that are null homologous in D', hence for which Int -y is a subregion of D'. Theorem. D
= u-ye./,Int
Proof. Let 15 = DU RD. If w E D, then w E Int -y for every small circle -y E about w. If w E RD, then w lies in an open compact subset K of D'\D. Then Do := D U K is a subregion of D' (see A.3(1)). By 12.4.1(3), there exists a cycle -y in D = Do\K such that K C Int C Do. In particular, -y E Y. and w E Int -y. For the apposite inclusion, let w E Int 'y with -y E . 7. In order to show that w E D, we may assume that w (1 D. Then, since 1-yI C D, we have Z ni-y i = 0 for the component Z of D'\D containing w. The index function incl.), is now well defined and constant on Z. Since w E Int -y, it follows that Z c Int -y. Hence Z C (Int -y Ul-yl)\D. Thus Z is compact. It follows that Z C RD and therefore that w E RD C D. Corollary (Behnke-Stein theorem). A pair D, D' of regions with D C D' is Runge if and only if every cycle -y in D that is null homologous in D' is already null homologous in D.
298
13. Runge Theory for Regions
Proof. D, D' is a Runge pair if and only if D = hit D for all -y
15,
i.e. if and only if
In the theorein and the corollary, we must admit cycles, not just closed paths. If D Cx \OE, D' Cx then D = C x but Int -y C D for every path -y G All cycles -y i + -y2 consisting of oppositely oriented circles -y 1 in E x and 72 E C\E about 0 are null homologous in D' but not in D. If D is a domain G, we can get by with paths -y G .r/ in the theorem; every cycle in G an then be turned into a closed path by joining the components of the cycle by paths in G
Historical note. H. Behnke and K. Stein proved their theorem in 1943 in [BS1 for arbitrary noncompact Riemann surfaces; they state the homology condition in such a way that D is simply connected relative to D' (loc. cit., pp. 411 445).
4. Runge regions. A region D in C is called a Runge region if every function holomorphic in D can be approximated compactly in D by polynomials. Since C[z] is dense in 0(C), D is a Runge region if and only if D, C is a Runge pair. The statements of Theorem 1 and the Behnke-Stein t heorem make possible a simple characterization of Runge regions. Theorem. The following statements about a region D in C are equivalent:
i) I) has no holes. D is a Runge region. iii) There is no open compact set 0 in the space C\D. iv) D is homologically simply connected. y) Every component of D is a simply connected domain. iii) = iv) i), with Proof. i) ii) iii) i). This is Theorem 1, i) D' = C. ii) 4=> iv). This is the Behnke-Stein theorem with D' = C. since all cycles are mill homologous in C. y). Since D is homologically simply connected if and only if eviv) ix).0 ery component of D is, the assertion follows from Theorem 8.2.6, i) The equivalence i) v) says in particular that the simply connected domains in lik2 are precisely the domains without holes; see 1.4. The next st at ement follows immediately. A bounded domain G in C ls simply connected if and only if C\G is eon n ee t ed.
There are, however, unbounded simply connected domains G in C for which C\G has (in)finitely many components (the reader should sketch examples). Wit h the aid of i) NO, the injection theorem 8.2.2 can be generalized:
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299
For every domain G in C whose complement C\G has a component with more than one point, there exists a holomorphic injection G —+ E. C is a domain Proof. If C\G has an 'unbounded component Z, then C\ Z (!) without holes' and can therefore he mapped biholomorphically onto E. Hence G C C\ Z can be mapped biholomorphically and injectively into E. Now let M be a component of C\G that has more than one point. Then z 1-3 1/(z — a), a E M, maps C\M biholomorphically onto a domain G 1 one unbounded component. Thus G 1 ad-whosecmplnta GI. 0 mits a holomorphic injection into E; hence so does G C CVV.f 5*. Approximation and holomorphic extendibility. If D, D' is not a Runge pair, then not all functions in 0(D) can be approximated in D by functions in 0(D'). We prove:
Theorem. The following statements about a function f E 0(D) are equivalent:
i) f can be approximated compactly in D by functions in 0(D'). ii) There exists a single-valued holomorphic extension f of f to the Runge hull D. Proof. ii) i). Clear, since f ID = f and D, D' is a Runge pair by Theorem 2 i) ii). We define :7 as in Subsection 3 and consider, in D', all the Let a pair Dy , fy with fy E 0(Dy ), subregions ay = D U Int 7, -y E E 5°, be called an f -extension if f-y ID = f. We claim: a) For every 'y E there exists an f -extension. h) If D-r , fy and Db, h are f -extensions, then f.), = fb on Dl ni3,. It follows immediately from a) and b) that there exists exactly one holomorphic function f on Ulcs D.-1 such that f 1.171. = ty , -y E S. Theorem 3 then implies ii). For the proof of a), we choose a sequence gn E 0(D') that converges compactly to f in D. Then, since urn If —g 7,11 = 0, gn is a Cauchy sequence Since Int .7 is a bounded region with 0(Int -y) C on 1-yl for every 7 E the maximum principle implies that gn is also a Cauchy sequence in Int -y ),); then Dy , f-y is an f-extension. := lim„,(g„la and hence in Dy . Let For the proof of b), it suffices (by the identity theorem) to prove that every component Z of (Int -y) n (Int 6) intersects the region D. But this is clear since
aZ C 0(Int 7) U 0(Int 6) c 1-y1 u 161 C
D.
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13. Runge Theory for Regions
§3. Holomorphically Convex Hulls and Runge Pairs If M is a subset of a region D in C, the set MD
:= {z E
D:
if (z)l < J IM
for all f E 0(D)}
is called the holomorphically convex hull of M in D. This is often written M instead of MD. For the circle S := 8E, we have S`c = E and ,§c. = S. The next statement follows immediately from the maximum principle for bounded domains.
If V
is relatively compact and open in D, then (0)D D
Let a, f3,
V.
E , z = x + iy. Then, since le( z I < er if and only if ax < r, always lies in the intersection of all the closed half-planes containing M. This implies: 7'
R
The holomorphically convex hull M is contained in the linearly convex hull of M.
Runge pairs D, D' can be characterized analytically by the property that every compact set K c D has the same holomorphically convex hull in D and in D'; this equivalence is the main result of this final section on Runge theory. 1. Properties of the hull operator. The following properties are immediate from the definition:
(1) M is closed in D. Moreover, ftsj C itTP, and D C
C 1-14--Di •
(2) r E D\I-C/I if and only if there exists an h E 0(D) such that Ihim < 1 < Ih(c)iThe next property is important for what follows. (3) It is always true that d(M,C\D) = d(it71,C\D). If M is compact, so is
R R,
Proof. Since M C we need only prove that d(M,C\D) < d(1171, C\D). Let ( V D. Since (z E 0(D), we have lw - (1 -1 < suPliz-a -1 : z E MI for all w E It follows that d(R, () = inffiz - (I : z E MI < w (I for all w E hence d(M,() < d(117,() for all.( E C\D. —If M is compact, then is closed in C because 0 < d(M,C\D) = d(R,C\D). Since lies in the linearly convex hull of M, which is bounded if M is, is also bounded.
R
R; R
R.
R
§3. Holomorphically Convex Hulls and Runge Pairs
301
By (2), M --= MD if and only if, for every c E D\M, there exists an h E 0(D) such that lh(c)I > i hi m. Our next result thus follows directly from Theorem 12.2.3. Theorem. The following statements about a compact set K in D are equivalent:
i) K = k D. ii) The space D\K has no component that is relatively compact in D. iii) Every bounded component of C\K contains a hole of D. iv) The algebra 0(D) is dense in 0(K).
For the proof, it suffices to observe that a bounded component of C\K contains a hole of D if and only if it intersects C\D. 1:1 The equivalence i) iii) immediately yields a more precise form of 1.1(1):
For every compact set K in D, there exists a smallest compact set K1 D K in D, namely kn, such that every bounded component of C\K 1 contains a hole of D.
The implication i) = ii) can be generalized as follows.
(4) For every compact set K in D, ff D is the union of K and all the components of D\K that are relatively compact in D. Proof. Let A denote the union of all the components of D\K that are relatively compact in D; let B denote the union of all the remaining components of D\K . Since each component is a domain,
A and B are open in D, D\K = A U B , and A
n B = 0.
We set M := K U A. Then, by 12.2.3(1) and (1), M C RD C MD. Hence M = k D if M = MD. Since D\M = B is open, M is closed in D and therefore compact, since RD is compact by (3). By the definition of B, no component of D\M is relatively compact in D; hence the theorem implies D that M = MD. Thus passage from K to RD "fills in the holes of D\K in D." The purely topological description of holomorphically convex hulls given by (4) is occasionally used in the literature as the definition of RD; cf. [NI, PP. 112-113, and [FL], p. 204. By a simple argument, one can also show (cf. [N], pp. 112-113):
If K = k D, then C\K has only finitely many components. The concept of the holomorphically convex hull comes from the function theory of several variables; it was developed in the 1930s (cf. [BT], Chapter VI, and the appendix by O. Forster). For regions D in C', n > 1, the hull RD of a compact
302
13. Runge Theory for Regions
set K c D, which is defined word for word as above, is in general no longer compact. The following holds: A domain G in C, 1 1. For more on this, see the appendix by O. Forster to Chapter VI of [BT], where general Runge pairs of Stein spaces are considered.
Every set
M in C is assigned its polynornially convex hull := {Z E C: Ip(z)1< IPI m for all polynomials p } ;
we have M' = M. A region D in C is called polynomially convex if, for
every compact set K C D, the (compact) set K' lies in D. The theorem contains the following statement: A region in
C is Runge if and only if it is polynomially convex.
This equivalence also remains valid for domains in C'.
Appendix: On the Components of Locally Compact Spaces. ura-Bura's Theorem Theorem 2.1 says in particular that a difference D'\D has compact connected components if and only if it contains nonempty open compact sets. This is a theorem of set-theoretic topology (which does not appear in current. textbooks). We prove it here in a more general situation; cf. Theorem 2 and Corollary 2.1. — X always denotes a topological space.
1. Components. We consider connected subspaces of X; every one-point subspace {x}, x E X, is connected. If A„ z E .1, is a family of connected subspaces of X such that any two have a nonempty intersection, then the union U A, is connected. Hence the union of all the connected subspaces of X containing a fixed point is a maximal connected subspace of X. Every such subspace is called a component — more precisely, a connected component — of X.
(1) Distinct components of X are disjoint. Every component of X is closed in X. The last statement follows because the closure A of A is connected whenever A is. — In general, components of X are not open in X. (2) Every subspace of X that is both open and closed is the union of components of X. equipped with the relative topology. The Examples. 1) Let X := Q C components are the points of X; no component of X is open; there are no open compact sets 0 in X.
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13. Runge Theory for Regions
, 1/n, ...} C R, equipped with the relative 2) Let X := {0,1,1/2, topology. The components of X are the points of X; every component {0} is open in X. The component {0} is the intersection of all the open compact subsets of X containing it. 3) Let X be a Cantor set in [0,11. The components are the points of X; no component of X is open; there are open compact sets O in X. 4) For regions D C D' C C, the difference D'\D is locally compact. In the case D := D'\Cantor set, D'\D has uncountably many compact components, and no union of D with finitely many of these components is a region. The following was used in 1.2.
(3) If D' is a domain, then every component Z of D'\D intersects the boundary of D. Proof Let a E Z, let b E D, and let -y be a path in D' from a to b. On -y, there is a "first" point c E Since Z is connected and the subpa.th from a to c passes through D'\D, we have C Z; hence c E Z.
al:).
2. Existence of open compact sets. Every compact component A of a locally compact (Hausdorff,) space X has a neighborhood base in X consisting of open compact subsets of X.
This theorem was proved in 1941 by M. Sura-Bura for (bi)compact spaces; cf. [SB]. The theorem appears implicitly in N. Bourbaki; cf. [Bo], p. 205, corollary and its proof. The significance of the theorem for function theory was pointed out by R. B. Burckel; cf. [Bu]. We prove S. . ura-Bura's theorem in Subsection 4, but we first derive a few of its consequences here and in Subsection 3. Since open compact sets, by 1(2), are always the union of compact components, our first result is immediate. Corollary 1. A locally compact space X has compact components if and only if there exist nonempty open compact sets in X. The union of all the compact components of X coincides with the union of all the open compact subsets of X, and in particular is open in X.
This contains the equivalence statement i)
