CONFORMAL REPRESENTATION BY
C. CARATH£0DORY C, CARATHEODORY
CAMBRIDGE AT THE THE UNIVERSITY AT UNIVERSITY PRESS PRESS ...
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CONFORMAL REPRESENTATION BY
C. CARATH£0DORY C, CARATHEODORY
CAMBRIDGE AT THE THE UNIVERSITY AT UNIVERSITY PRESS PRESS 1969 1969
PUBLISHED PUBLISHED BY BY THE SYNDICS SYNDICS OP OF THE THE CAMBRIDGE CAMBRIDGE UNIVERSITY UNIVERSITY PRESS PRESS THE Bentley House, House, 200 200 Euston Euston Road, Road, London, London, N.W.l N.W.l Bentley American Branch: Branch: 32 32 East East 57th Street, New American New York, N.Y. 1002:!
Standard Book Book Number: 521 521 07628 55
First Edition Edition 1932 Second Second Edition 1952 Reprinted 1958 1963 1969
First First printed printed in in Great Great Britain Britainatatthe theUniversity UniversityPress, Press,Cambridge Cambridge Reprinted Reprinted in Great Great Britain by William William Lewis Lewis (Printers) Ltd., Ltd., Cardiff Cardiff
NOTE BY NOTE BY THE GENERAL GENERAL EDITOR EDITOR·
pP
ROFESSOR Caratheodory Carathéodory made made the few few corrections corrections necessary necessary in of the the first first edition edition of of this this tract, tract, completed completed the manuscript the text of of the new Chapter VIII, VIII, compiled compiled the the Bibliographical Bibliographical Notes Notes afresh, and and wrote the the Preface Preface to to the thesecond second edition, edition, during during the the later latermonths monthsof of1949. 1949. At his request, Mr U. E. E. H. Reuter, Mr G. Reuter, of of the theUniversity UniversityofofManchester, Manchester, agreed to revise revise the the author's author'sEnglish. English.This Thiswork workwas wascompleted completed shortly shortly before Professor Caratheodory's Carathéodory's death death on on 2 February 1950. 1950. Mr Mr Reuter before Professor then kindly the task of reading reading the the proofs and attended attended to kindly undertook undertook the proofs and all points of detail which arose while whilethe the tract tract was through the was going going through which arose Press.
W.V.D.H. CAMBRIDGE January January1951 1951
PREFACE TO TO SECOND SECOND EDITION
T T HIS reprint of my tract is almost without change, save for the ad
HIS reprint of my tract is almost without change, save for the adchapter on dition of a chapter on the the celebrated celebrated theorem theorem of of Poincaré Poincare and Koebe Koebe on uniformisation. uniformisation. If If II have havesucceeded succeeded in in making making this chapter rather because I have have been been able to to avail avail myself myself of the beautiful beautiful short, it is because short, of van van der Waerden (30), the proof of (30), which which has enabled me to expound expound the topological side of the problem problem in in aa few few pages. pages. topological
C. CARATHEODORy C. CARATHEOOORY MUNICH :MUNICH
December 1949
PREFACE TO EDITION PREFACE TO THE THE FIRST EDITION little book is the outcome of lectures which I have given at T various times and at different places (Gottingen, Berlin, Athens, T HIS
HIS little book is the outcome of lectures which I have given at various times and at different places (Gottingen, Berlin, Athens, contains the theory theory of Munich, Munich,and andat at the the University Universityof ofHarvard). Harvard). It It contains conformal representation representation as as itit has has developed developed during during the the last last two two decades. decades. The first half of the book book deals deals with with some some elementary elementary subjects, subjects, knowknowThe ledge of the the general general theory. ledge of of which which isis essential essential for for the understanding of The exposition exposition of of this this theory theory in in the the last three chapters uses the simplest The today. methods available today. The original original manuscript, manuscript, written written in in German, German, has has been been translated translated by The l\1. Wilson of the University University of of Liverpool Liverpool and by by Miss Miss Margaret Margaret Mr B. M. Kennedy Newnham College. College. II wish express here here my my warmest warmest Kennedy of Newnham wish to express thanks for the care care they they have taken that that the the most most intricate intricate arguments should be made clear to the the reader. reader. IIam amalso also indebted indebted to to Prof. Prof. Erhard Erhard (Berlin) and andtotoProf. Prof.Tibor TiborRadó Rado(Columbus, (Columbus,Ohio) Ohio) for for various various Schmidt (Berlin) improvements in the mathematical demonstrations, and and to to Miss Miss Kennedy mathematical demonstrations, Kennedy for several several suggestions suggestionsthat that simplified simplifiedthe thetext. text. Finally, my thanks are for due to the staff Press for for the admirable staff of of the the Cambridge Cambridge University University Press admirable due way in which way which their part of of the the work work has has been been carried carried out. out. C. CARATHEODORY CARATHEODORY C. ATHENS December 1931
CONTENTS PAGE
PREFACE
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INTRODUCTION. HISTORICAL INTRODUCTION. HISTORICAL SUJOIARY .
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1 1
CHAP.
MOBIUS TRANSFORMATION TRANSFORMATION I. MOBIUS . . . . § 5. Conformal Conformal representation in general general . . . . §§ 6—9. 69. Mobius Mobius Transformation Transformation . §§ 1012. 10—12.Inva.ria.nce Invariance of ofthe the crossratio crossratio .. . . . 13—15.Pencils Pencils of circles . . . . . §§ 1315. . . . . . 16—22.Bundles Bundles of circles .. §§ 1622. . §§ 23—25. 2325. Inversion Inversionwith with respect respect to to aa circle . . . §§ 26—30. 2630. Geometry Geometryof of.Mobius Mobius Transformations Transformations . . II. II. NONEUCLIDEAN NONEUCLIDEAN GEOMETRY GEOMETRY 31—34.ln¥ersion Inversionwith withrespect respect to to the circles of a bundle . §§ 3134. § 35. Representation of of a circular area on itself . 35. Representation . . . . . . . 37. NonEuclidean §§ 36, 36, 37. NonEuclidean Geometry Geometry Angle and and distance distance . . . . §§ 38—41. 3841. Angle . . . . . 42. The § 42. The triangle triangle theorem theorem . . § 43. NonEuclidean length length of of a curve curve . 43. NonEucidean . . . 44. Geodesic . . . . . . § 44. Geodesic curvature §§ 4547. 4547. NonEuclidean NonEuclidean motions motions . . . . . . §~ 48. 48. Parallel . . . . . . Parallel curves curves IlL ELEMENTARY ELEMENTARYTRANSFORMATIONS TRANSFORMATIONS III. Theexponential exponential function function . . . . §§ 49—5 4951. 1.The §~ 52, 52, 53. 53. Representation Revresentation of of a rectilinear rectilinear strip stripon on aa circle circle . § 54. . . 54. Representation Representation of of aa. circular circular crescent crescent . . Representation of Riemann surfaces . . §§ 55—59. 5559. Representation surfaces . 60, 61. 61. Representation §§ 60, Representation of of the the exterior exterior of of an ellipse ellipse . . 62—66.Representation Representationofof an an arbitrary §§ 6266. arbitrarysimplyconnected simplyconnected . . . . bounded domain domain domain on a bounded . SCHWARZ'S LEMMA LEMMA IV. SCHWARZ'S . . . . . § 67. 67. Schwarz's Schwarz's Theorem Theorem . 68. Theorem § 68. Thuorem of of uniqueness uniqueness for the the conformal conformal representation . . . . of simplyconnected domains . . . . . . 69. Liouville's . § 69. Liouville's Theorem Invariant enunciation of §§ 70—73. 7073. Invariant of Schwarz's Schwarz's Lemma . § 74. 74. Functions Functions with positive real parts . . . . 75. Harnack's . . . . . . . § 75. Harnack's Theorem Theorem . 76. Functions real parts . . . § 76. Functions with with bounded bounded real with algebraic 77—79. Surfaces §§ 7779. Surfaces with algebraic and logarithmic logarithmic branchbranchpoints .. . . . . . . . . . .
3 4 5 5 7 8 11 11 13 13
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16 16 17 18 18 19 19 21 22 22
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23 25 26 27 27 28 29
31 32 39
40 40
41 43 44 45 45
viii
CONTENTS
CHAP. CUAP.
IV.
PA.GE PAGE 80—82. Representation of simple domains . domains.• §§ 8082. . §§ 8385. Representation upon one another of domains domains concon. taining circular areas .. . . . . . . . . . . . . . §§ 86. )'roblem . §§ 87, 88. Extensions of ofSchwarz's Schwarz's Lemma . 88. Extensions . . . . . . . . §§ 89—93. 8993. Julia's Theorem Theorem .
V. THE FUNDAtdENTAL FUNDAMENTAL THEOREMS THEOREMS OF OFCONFORMAL CONFORMAL V. THE REPRESENTATION . . . . . §§ 94. Continuous Continuous convergence convergence 96. Limiting . . . . . §§ 95, 96. Limiting oscillation oscillation §§ 97—99. 9799. Normal families of bounded bounded functions . . . Existence of of the the solution solution in in certain problems §§ 100. 100. Existence problems of the . . . . . . . calculus of variations .. 101—103. Normal families families of regular regular analytic functions . ~§ 101103. 104. Application Application to conformal conformal representation representation . . §§ 104. . 105—118. The main theorem of conformal conformal representation representation . §§ 105118. §§ 119. Normal Normal families families composed composed of of functions functions which which transform simple simple domains domains into into circles circles.• . . . . form . 120—123. The kernel of a sequence of domains domains . §§ 120123. . . . . . . . §§ 124. Examples . . 125—130. Simultaneous §§ 125130. Simultaneous conformal conformal transformation of dodotransformation of each within another . . . mains lying each . .
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VI. TRANSFORMATION OF THE FRONTIER VI. TRANSFORMATION OF FRONTIER . . §§ 131—133. 131133. An inequality due to Lindelof Lindelof .. 135. Lemma . §§ 134, 134, 135. Lemma 1, on representation of the frontier 136. 2 . . . . . . . . ~ 136. Lemma . § §§ 137, 138. domain into into 138. Transformation Transformationofof one one Jordan Jordan domain another . . . . . . . . . §§ 139, 140. an analytic analytic curve curve 140. Inversion with respect to an . . . . 141—145. The inversion principle . M 141145. principle 146—151.Transformation Transformation of corners corners .. . . . ~§ 146151. 153. Conformal transformation on the frontier .. §§ 152, 153. Conformal transformation .
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VII. TRANSFORMATION TRANSFORMATION OF OF CLOSED CLOSED SURFACES SURFACES . . . . . 155. Blending §§ 154, 155. Blending of domains .. Conformal transformation transformation of of aa threedimensional §§ 156. Conformal threedimensional surface . . . . . . . . . . Conformal representation representation of aa. closed §§ 157—161. 157161. Conformal closed surface on . . . . . . . . . . a sphere .
46
50 52 52 52 53
58 58 61 61 62 63 66 66 73 74 77
77 81 82 84
85 87 88 91 96 98 99 100
ix
CONTENTS CHAP.
PAGIC PAOE
VIII. THE THEGENERAL GENERALTHEOREM THEOREMOF OFUNIFORM UNIFORAIISATION ISATION
163, 164. 164. Abstract surfaces . . §§ 162, 163, ::~urface11 166. The ~~ 165, 166. The universal universal covering covering surface surface . § 167. 167. Domains and their their boundaries boundaries .. 168. The ~ 168. The Theorem Theorem of of van van der Waurden Wa.crdeu 169. Riemanu . . . ~ 169. Riemann surfaces surfaces ~§ 170, Uniformisation Theorem 170, 171. 171. The Uniformisation Conformal representation representation of of aa torus §§ 172. 172. Uonformal
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BIBLIOGRAPHICAL NOTES NOTES .. BIBLIOGRAPHICAL
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103 104 105 106 108 108 110 111 113
INTRODUCTION HISTORICAL SUMMARY SUMMARY
of two two areas areas on 1. By By an anisogonal isogonal (winkeltreu) (winkeltreu) representation representation of on one one continuous, and and continuously differentiable differentiable another we mean a oneone, oneone, continuous, representation of the areas, areas, which which is such such that thattwo two curves curves of of the the first first representation which intersect an angle angle ot are transformed into two two curves curves area which intersect at at an are transformed intersecting at the the same same angle angle ot. If the sense sense of rotation of a tangent tangent is preserved, an isogonal isogonal transformation transformation isis called called conformal. conformal. Disregarding as trivial trivial the Euclidean magnification (.Ahnlichkeitstmns(Ahnlichkeitstransformation) say that that the oldest known transformation formatwn) of of the the plane, plane, we may say known transformation of this kind kind isisthe thestereographic stereogmphicprojection projection of of the thesphere, sphere, which which was was used used by Ptolemy (flourished (flourishcdin inthe thesecond secondquarter quarter of ofthe the second secondcentury; century; died after A.D. 161)for forthe the representation representationof of the the celestial celestialsphere; sphere; it transA.D. 161) the sphere sphere conformally conformally into a plane. A A quite quitedifferent differentconformal conformal forms the forms representation of the the sphere representation of sphe~e on on a plane plane area area isis given given by by Mercator's .bIercator's Projection; this the the spherical spherical earth, along a meridian meridian circle, circle, is Projection; in this earth, cut along conformally representedon onaaplane planestrip. strip. The first map constructed by conformally represented (1512—1594)in in 1568, this transformation transformation was was published by .biercator (15121594) adopted for and the method method has has been been universally universally adopted for the construction construction of seamaps. 2. AAcomparison comparison of of two two maps maps of of the thesame samecountry, country, one one constructed constructed by stereographic projectionofofthe the spherical sphericalearth earthand and the the other other by by stereographic projection Mercator's Projection, will will show conformal transformation does not show that that conformal similarity of ofcorresponding correspondingfigures. figures. Other nontrivial nontrivial conformal conformal imply similarity representations representations of a plane plane area area on on aa second second plane area are obtained by comparing various stereographic comparing the various stereographic projections projections of the the spherical spherical earth earth which which correspond correspond to different positions positions of the the centre centreof ofprojection projection on on the earth's earth's surface. surface. It was was considerations considerations such such as as these these which which led led Lagrange in 1779 1779 to to obtain all Lagrange (1736—1813) (17361813) in all conformal conformal representations representations of a portion of the the earth's earth's surface surface on on aa plane plane area area wherein wherein all circles of latitude and are represented represented by bycircular circulararcs(I). arcsl. and of longitude longitude are 3. In 1822 Gauss (1777—1855) (17771855) stated completely solved solved the 1822 Gauss stated and and completely general problem of of finding finding all all conformal conformal transformations transformations which which transtransform a sufficiently small neighbourhood neighbourhoodofofaa point point on an arbitrary form sufficiently small arbitrary
2
INTRODUCTION
analytic surface surface into intoaaplane planearea area(2). work of Gauss appeared to (2). This work give whole inquiry final solution; solution; actually it it left left unanswered unanswered give the whole inquiry its final the much much more more difficult difficult question question whether whether and and in in what what way way a given finite portion surface can represented on portion of plane. portion of of the surface can be be represented on aa portion of the plane. Dissertation This was first pointed pointedout outbybyRiernann Riemann(1826—1866), (18261866), whose whose Dissertation {1851) in the the history hiBtory of ofthe theproblem problem which which has (1851) marks marks aa turningpoint in been decisive decisive for for its its whole later development; development; Riemann not only whole later only introintroduced all ideas which which have have been been at the the basis basis of of all all subsequent subsequent duced all the ideas investigation investigation of the problem of conformal conformal representation, representation, but also showed that the theproblem problem itself itself isis of offundamental fundamental importance importance for for the theory theory of of functions (3). functions(3).
4. Riemann Riemann enunciated, enunciated, among among other results, results, the the theorem theorem that that every every simplyconnected plane area which whole plane simplyconnected plane which does not not comprise comprise the whole can be represented conformally conformallyon onthe the interior interior of ofaa circle. circle. In the proof of this theorem, forms the the foundation theorem, which which forms foundation of the whole whole theory, theory, he assumes assumes as as obvious obvious that that a certain problem problem in the calculus calculus of variations possesses aa solution, and possesses and this thisassumption, assumption,asasWeierstrass JVeit!rstrass(1815—1897) (18151897) first invalidates his proof. proof. Quite Quite simple, simple, analytic, analytic, and in first pointed out, invalidates every way way regular regular problems problems in in the thecalculus calculusofofvariations variationsare arenow nowknown known which do not not always always possess possess solutions(4). solutions(4). Nevertheless, Nevertheless, about about fifty fifty years years after Riemann, rigorouslythat that the particular Riemann, Hubert Hilbert was was able to prove rigorously problem which arose arose in in Riemann's solution; this problem which Riemann's work work does does possess possess aa solution; theorem theorem is known known as as Dirichlet's Dirichiet's Principle(5). Principle5). Meanwhile, however, however,the the truth truth of Meanwhile, of Riemann's Riemann's conclusions conclusions had been been established manner by by C. C Neumann and, established in a rigorous rigorous manner and, in in particular, pa~ticular, by by H. A. A. Schwarz6. Schwarz(6).The Thetheory theorywhich whichSchwarz Schwarz created created for for this this purpose purpose however, somesomeis particularly elegant, interesting and instructive; instructive; ititis,is,however, what intricate, intricate, and and uses uses a.a number the theory theory of of the what number of of theorems theorems from from the which must must be be included included in any complete logarithmic potential, proofs of which complete account of the the method. the work of a account of method. During During the present present century century the work of mathematicians has number of mathematicians has created creatednew new methods methods which which make possible simple treatment treatment of our problem; ititisis the a very simple thepurpose purpose of of the the following following pages to give give an an account which, while while as as short short as pages to account of these methods methods which, possible, shall yet be essentially complete. complete. possible,
CHAPTER CHAPTER I
MOBIUS TRANSFORMATION MOBIUS 5. Conformal representation in in general. general. 5. Conformal It is is known known from from the theory theory of functions functions that an an analytic analytic function function
w =f(z), which w=f(z), which is is regular regular and and has hasa anonzero nonzerodifferential differentialcoefficient coefficient at the point point zz = z0, z0 , gives a continuous oneone oneone representation representation of a certain neighbourhood of the point neighbourhood of point z0 z 0 of the zplane zplane on on aa neighbourhood neighbourhood of a point point w0 u·o of the wplane. wplane. Expansion of the the functionf(z) function f(z) gives Expansion of gives the the series series wuo=A(zzo)+B(zzof+ ····} A =/=0; A#0;
and ifif we we write
...... (51) (5"1)
...... (5"2) A= w — w0= where positive, then (5·1) where t,t, A, X,and anduuare arereal, real,and and r,r, a, a, and p are positive, (51) may be written . pe"' pe'" = are•.+t) { 1 + (r, t)},} t)}, 5•3 ...... (5"3) . ( ) 1lim cfr(r,t)=0. 1m r,t =0. ,._o z — :0
= re",
0
two relations: This relation is equivalent to the the following following two
u=A+t+$(r,t), p=ar{1 +x(r,t)}, U=A+t+fl(r,t),} p=ar{l+<X(r,t)}, . lim 0, urn <X (r, t) ==0,
ro
1urn 1m ,6 {1 (r, t) = 0. 0.1 ro
(5.4 . ..... (5·4)
When r == 0 relations becomes becomes 0 the second of these relations
u=A+t, U=A+t,
(55) ...... (5"5)
and expresses between the the direction direction of of aa curve curve at at the expresses the connection connection between point z 0 and the the direction direction of ofthe thecorresponding corresponding curve curve at at the the point pointw0. w0 • Equation (5·5) shows shows in representation furnished by in particular particular that that the representation w =1(z)at at the the point point;z 0isis isogonal. the function function w=f(z) isogonal. Since Since the the derivative derivative that the f' no zeros zeros in in aa certain certain neighbourhood neighbourhood of z0 , it follows follows that 1' (z) has no byf(z) on a portion of representation effected effected by f(z) of ofaa neighbourhood neighbourhood of of zo on thewplane the wplane is is not not only only continuous continuous but butalso alsoconformal. conformal.
The first The first of of the the relations rela.tions (54) (5·J) can can he beexpressed expressed by by saying saying that that "infinitely small" circles circles of of the the zplane zplane are are transformed transformed into into infinitely infinitely small circles of the the u'plane. wplane. Nontrivial conformal conformal transformations transformations small circles exist however for which finite circles; circles; these however for which this is also also true of Qf finite these transtransformations will be investigated first .. formations will be
4
MOBIUS TRANSFORMATION TRANSFORMATION MOBIUS
[CHAP. [CHAP. II
6. Mdbius Mobius Traniformation. Transformation. 6. Let A, A, B, B, CCdenote denote three three real real or orcomplex complex constants, constants, A, B, C 0 their their· conjugates, and x, :x, x a complex complex variable conjugates, variableand and its its conjugate; conjugate; then then the equation — ...... (6"1) represents circle or or straight straight line line provided provided that represents aa real circle . ..... (6"2) BB>(A +A)(CiC). (62) BB> (A+ A)(C+ C). Converselyevery everyreal realcircle circleand andevery everyreal realstraight straight line can, by by suitable suitable Conversely choice of choice of the the constants, constants, be be represented represented by by an an equation equation of the the form form (6"1) (61) satisfying condition (6"2). (62). If now in (61) satisfying condition now in (6"1) we we make make any of the the subsubstitutions ...... (6"3) (63) y=:x+A, ...... (64) (6"4) '!! = p.a:,
y=:x·1 1
or
...... (6·5)
the equation equation obtained obtained can be be brought brought again again into into the the form form (61), (6"1), with with new 0 which which still still satisfy satisfy condition condition (6"2). The substitunew constants constants A, B, C transforms those lines (61) (6"1) for for which which tion {6"5) transforms those circles circles and and straight straight lines (6 C+ 0 + 0 = 0, i.e. i.e. those which which pass pass through through the the point point x.x== 0, into straight straight lines; we lines; we shall shall therefore therefore regard regard straight lines lines as as circles circles which which pass pass point x:x = oo . through the point
7. If we we perform perform successively successively any number of of transformations transformations (6"3), any number (64), {6"5), taking each (6"4), each time arbitrary arbitrary values values for for the the constants constants A, A, p., the resulting transformation is always always of the form rt.:x + f3 (71) y= y:x+B;
...... (7"1)
here; here ~./3,{3,y,y,6 I)are areconstants constantswhich whichnecessarily necessarily satisfy satisfy the the condition condition ~/) {3y =I= 0, ...... (72) (7"2) since would be either constant since otherwise otherwise the righthand member of (7"1) (71) would or meaningless, meaningless, and and (7"1) (71) would would not give give aa transformation of the :xplane. Conversely, any bilinear bilinear transformation transformation (7"1) (71) can Conversely, any can easily easily be be obtained obtained by by (71) also means of transformations hence (7"1) also transtransmeans transformations (6·3), (6·4), (64), (6·5), and hence forms circles. forms circles into circles. (7"1)was wasfirst firststudied studied :fl'liibius (17901868), The transformation transformation(71) byby Möbius (7)(7)(1790—1868), and will therefore be called Tran.iformation. called Miibius' Möbius' Trantformation. 8. The Thetransformation transformation inverse inverse to (71), (7"1), namely namely :r = '8y +
yy7y —
f3,
~
( '8) ( oc) {3y =I= 0,
...... (81) (8"1)
§§ 610]
GENERAL GENERAL PROPERTIES PROPERTIES
5
transformation. Further, Further, if we perform perform first first the the transis also also a MUbius' Mobius' transformation. formation (7"1) and then thenaasecond second Möbius' Mobius' transformation transformation formation (71) and
p, z = a.,y+ ~. ~ ' y,y + o,
"" a a., o,  ,.., y,
+0,
result isis aathird thirdMUbius' Mobius' transformation transformation the result —
—
z=
Ax + B Ax+B l'x+A'
with nonvanishing determinant — BI' Br = (a.8 Py) (a.,~ p,y,}=f= 0. AAThus we we have have the theorem: the the aggregate aggregate of of all all Möbius' :bfiibius' transformagroup. tions frwms forms a(~ group.
9. Equations Equations (71) (7·1)and and(81) (8"1)show show that, that, ififthe thexplane xplaneisisclosed closed by by the the every Möbius' addition point xa_·= Miibius' transformation traniformation isisaaoneone on~one addition of the point = ~, et'M"!J of the the closed closed .rplane transformation of xplaneinto intoitself. itself.If If7y =f= 0, the point pointyy=a.Jy = toz=—8/y; but ifify=Othe corresponds to the point x ~ oo, andy= and y = oo to x = 8/y; but y = 0 the points x = oo and y == oo correspond to each other. From (7"1) we we obtain
(ly_ a6—$y
dx (yx+S)2' 5, the the representation is conformal conformal except except at at the points x = oo so that, by so by§ 5, and x = = — ofy. In order order that these these two two points points may may cease cease to be be exexceptional we we now now extend the the definition definition of conformal conformal representation as ceptional follows: =f(x) will will be be said said to to transform transform the the neighbourhood neighbourhood follows: aa functiony =f(x) o' if the function x 0conformally conformally into aa neighbourhood neighbourhood of of y = oo of a point z, 1/f(z) transforms TJ = = 1//(x) transforms the neighbourhood neighbourhood of ofx0 x 0 conformally into a neigh=0; bourhood of of"'= 0; also alsoyy=f(x) = f(x)will willbe besaid saidto totransform transform the the neighbourneighbourhood of x = oo conformally iiito into aa neighbourhood neighbourhoodofofYo '!/o if if '!/y=4'(E)=f(l/E) = cp (~) =/(1/~)
E =00 conformally into into aa neighbourhood transforms the neighbourhood neighbourhood of of~= of '!/o· In this may have have the value this definition definition y0 '!/o may value oo. In virtue of of the the above above extensions extensions we we now now have the the theorem: theorem: every erery Möbius' transformation conformal representation representation qf :b:liibius' transformation gives gives a oneone oneone conformal qf the the entire closed on the entire enti1·e closed .vplane xplane on entire closed closed yplane.
10. Invariance 10. In variance of of the the crossratio. crossratio. x3, xx4 denote any aiiy four four points of Let x1, x 11 x2, x 2 , :ra, of the thexplane, xplane,and andy,, y,3/2, y2 , 4 denote y 3 , yy4 which correspond correspond to them py the Möbius' Mobius' transformatransforma4 the points which tion (71). in the first (7"1). If we we suppose suppose in first place that all all the the numbers numbers :r,, y,
____
6
MOBIUS TRANSFORMATION MOBIUS TRANSFORMATION
[CHAP. I (CHAP.
finite, we have, for for any two of the points, are finite, we have, — / rx.x,. + {J rt.:l:; + {J rxO  /Jy i'k !/i = y.x,. + 8  y.x; + o — '!/k= < y.xk + 8)( y.xi + o) (.xk.x;), yZk+8
and consequently, consequently, for for all four, (yi Y•) (Ys '!/2) _(.xi(x, — :r.)(.xa— X2) ...... 'lO•l (10"1) (J/1~Jh)(!j;= Y•) (.xi :x2)(.xa .X4) · The expression 'fhe expressrnn x4) (.x,(a,1 — x,) (:xi(x3 — :x2) z2) (x1 — x2) (z3 — x4) (:xi.x2)(.xax•) is called the crossratio of the four points points x,, .xi,z2, .x2,x3, .x3 , :r4 , s~ that, that, by by (10 (1 0·11), ), the crossratio crossratio is Mobius' transformation. is invariant under any Möbius' calculation shows shows that thatequation equation(1(10"1), when suitably suitablymodimodiA similar calculation 01), when fied, isis still still true ifif one fied, one of of the thenumbers numbers x,.X;or orone one of of the thenumbers numbers y, y 1 is is if, for example, :r2 = oo and y, '!/I = oo , infinite; if, y,'!/2  :ri  :x, 
YsY4
...... (10"2)
be two 11. Let z1, .xi> x2, :r2 , x3 .x3 and Y~> y 2, yy33 be two sets each each containing containing three unequal unequal complex complex numbers. numbers. We We will will suppose supposeinin the the first first place place that that all all six numbers numbers are are finite. finite. The equation {JJI '!/HYs '!/2) _ (:ri :x) (:ra .X2)
<J/IY2)(yaY) (.r1—z2)(x3—x) (:xi.x2)(:xa.x) (y1—y2)(y3—y)
...... (111) (11•1)
when solved for for yp yields aa Mobius' as is easily when solved Mobius' transformation transformation which, which, as easily verified, the corresponding corresponding point y 1*, and verified, transforms transforms each each point .x1 into the 10 now now shows theonly onlyMöbius' Mobius'transformation transformationwhich which does does so. shows that that ititisisthe §§ 10 This result remains remains valid validwhen when one one of ofthe thenumbers numbersz1 :x1 or or yj y1 is is infinite, provided of of course course that that equation (11 (11"1) is suitably suitably modified. modified. provided 12. Since Since aa circle circle isis uniquely uniquely determined determined by three points points on on its circircumference, Mobius' transformations transformations which which cumference, §§11 11may maybe be applied applied to to find Möbius' second given given circle circleor orstraight straight line. line. Thus, transform a given circle into a second Thus, = —11 and '!/1 = for = 1, for example, example, by taking :xi = = 1, 2'2 :r2= i, :x3 == 0, 'y2 !/2 = 1, p3 '!Ia = oo , we obtain obtain the transformation .1—z .1.x y=l1+.x'
...... (12•1)
i.e. one of the transformations which .vI== 11 on i.e. which represent the the circle circle l.x on the the real axis, < 1of ofthe theunitcircle unitcircle on on the the upper uppe1· half half real axis,and andthe theinterior interiorl.xx I
common diameter of anl B1 common of A1 A 1 ani B 1 which which cuts C1 0 1 (the circle into into which which C0
is transformed) orthogonallyisisaacircle circleof ofthe the plane plane cutting all three is transformed) orthogonally three Hence a circle cuts all AI> B1, B 1, 01 orthogonally. orthogonally. Hence circle exists which which cuts circles A1, B, C0 orthogonally. orthogonally. three circles A, B, Secondly, Secondly,ifif AA and and B B touch, there there isisaa.Möbius' Mobius'transformation transformationwhich which lines, and C 0 into into aa circle circle C1. 01 • transforms transforms them them into two parallel straight lines, Since 0C1 has one one diameter diameter perpendicular perpendiculartoto the the two two parallel parallel straight Since 1 ha.s lines, a.a circle existsininthis thiscase casealso alsocutting cutting all all three three circles circlesA, A, B, B, 0C circle exists lines, orthogonally. Finally, if A and and B B have have two two points points in common, common, there Mobius' Finally, there is a Möbius' transformation straight lines intersecting intersecting transformation which transforms them into two straight at which does doesnot notpass passthrough through0.0. Two at aa point point0,0,and andC0into intoaacircle circle C1 0 1 which Two cases liesoutcide outside the thecircle circle C1 0 1 there is is cases must must now nowhe be distinguished: distinguished: ifif 00 lies
r
Fig. 22
Fig. 33
again a circle if 0 lies again circle cutting cutting A,, A 1 , B1, B 1 , and and C1 0 1 orthogonally; orthogonally; whereas whereas if lies inside 0Cl1 there there is is a circle inside circle rr such such that thateach eachofofthe thecircles circlesA1, AI>B1, BI>C1 01 intersects rat F at the intersects the extremities extremities of of a diameter diameter of of F. r. We have thus proved the following following theorem: theorem: any anythree three coplanar circles satisfy at must satiify atleast leastone one of ofthe the following following conditions: conditions: the thethree threecircles circles have orthogonal circle K, or a common orthogonal m· they they pass pass through through aacommon common point, or or they can be they be transformed tmniformed by by aa Ziföbius' Mobius' transformation into into three three circles circles which cut cutaafixed fixedcircler circle rat at the theextremities extremitiesqf ofaadiameter diameterofofr.r. Itltfollows which follows readily from the the proof given t.hat that if the rea.dily from proof given the three t.hree circles circles A, B, C0 do do not not belong to the the same same pencil pencil the the circle circleKK is is unique; belong to unique; further, further, it will will be be proved below belowthat that three given of the the proved given circles circles cannot satisfy satiify more than one one of three conditions enumerated.
10
MOBrUS TRANSFORMATION MOBIUS TRANSFORMATION
(cHAP. [CHAP. I
17. We now define types of offamilies families of ofcircles circles which we call bundles define three types of of circles. An elliptic elliptic bundle bundle of of circles circlesconsists consists of of all all circles circles of of the plane which cut aa fixed circlerTatat the the extremitie~ extremities of of aa diameter diameterof ofr.r. The fixed circle The circle circle itselfbelongs belongs to to the the bundle bundle and and isiscalled called the theequator equator of of the the bundle. bundle. r itself A parabolic bundle bundle qf of circles circles consists plane which which A consists of all circles of the plane pass through throngh aa fixed fixed point, point, the thecommon common point of the bundle. bundle. A hypprbolic bundle plane which which A hyperbolic bundleof ofcircles circlesconsists consists of of all all circles circles of the plane cut cut aa fixed fixed circle circle or or straight straight line line orthogonally. orthogonally. These three figures distinct:: every 'fhese figures are essentially essentially distinct every pair pairof ofcircles circles of of an elliptic bundle intersect at at two two points; points; every everypair pairof ofcircles circles of a parabolic parabolic bundle either either intersect intersect at at two two points pointsor ortouch touchone oneanother; another; but but aa hyhybundle perbolic pairs of of circles circles which which have have no no common common point. point. perbolic bundle contains pairs 18. Bundles Bundles of ofcircles circlesnevertheless nevertheless possess possess very very remarkable remarkable common common properties. properties. For example: example: if if A, A, B B are aretwo twocircles circles of ofaabundle, bundlP, all all the the circles of the the pencil pencil which which coatains cofltainsA, A, BB belo11g belongtotothis this bundle. bundle. For a circles qf parabolic bundle of this this theorem theorem isis obvious; obvious; for for aahyperbolic hyperbolic parabolic bundle the the truth of follows from orthogonal circle bundle bundle it follows fromthe thefact factthat that the orthogonal circle of of the bundle bundle cuts the the circles circles A, B—and Band therefore therefore cuts cuts every every circle circle 2f pf the the pencil pencil containing and for containing A, B—orthogonally; Borthogonally; and for an elliptic elliptic bundle bundle ititfollows follows
from an elementary theorem from theorem of of Euclid. Euclid. The of the theoremisis equally equally simple: simple:ifif a plane The proof proof of the following following theorem plane containsaa bundle bundleof ofcircles circlesand andan an a1·bitrary arbitrary point point P, F, which, contains whick, if the tke bundle is parabolic, parabolic, does point of does not not coincide coincide with the tke common common point of the bundle, tken infinite number numbm· of ofcircles circles of oj the tke bundle, bundle, and andthese these circles then Plies P lies on an infinite through P form aa pencil. tkrougk pencil.
of a bundle do not belong 19. Let A, B, C G be be three circles circles of bundle which which do belong to and let let D D be fourth circle of the the bundle; the same same pencil, pencil, and be any fourth circle of bundle; then, starting with with A, A, B, B,CGwe wecan, can,bybysuccessive successiveconstruction constructionofofpencils, pencils, pencil of of circles circles which contains D, and all all of of whose whose members members arrive at aa pencil are circles circles of ofthe the bundle. bundle. For there is on on D at are at least least one one point point P which which neither a common is neither common point nor a limiting limiting point point of of either either of of the two pencils determined determined by by A, A, Band B and by by A, C' G and and which which does doesnot not lie lie on on A; A; we we can therefore draw through through PP two two circles circlesE, E,F, F, so so that E E belongs to the belongs to therefore draw pencil A, A, B, B, and and FF te E, FF are pencil tr the the pencil pencil A, C. G. The 'fhe circles circles E, are distinct, distinct, since A, B, C C do not not belong belong to the thesame samepencil, pencil, and andthe thesecond second theorem theorem now shows showsthat that D I) belongs to the the :pencil pencil determined determinedbyE, by E, F. of§ belongs to of § 18 now that a bundle of circles is is uniquely uniquely determined determined by by any any three three It follows follows that of its members which which do do not not belong belong to to the the same same pencil, pencil,and and in inparticular particular
17—23] §§1723]
INVERSION
11
that that three threecircles circles of of an elliptic elliptic bundle bundle which which do not belong belong to the the same same pencil pencil cannot cannot have have aa common commonorthogonal orthogonalcircle; circle;for forifif they they had had they would define was both elliptic elliptic and and hyperbolic. hyperbolic. would define aa bundle bundle that was
20. The Thecircles circles obtained obtained by Mobius' transformation by applying applying aa Möbius' transformationtoto all all the circles also form formaa bundle, bundle,and and the two circles of a bundle also two bundles bundles are of of the same kind. For parabolic and hyperbolic hyperbolicbundles bundlesthis this theorem theorem isis an immediate parabolic and immediate consequence of denote by by consequence ofthe the definitions definitions of of these these figures. figures. We therefore denote jf the aggregateofofcircles circlesobtained obtained from from the circles circles of a given given elliptic elliptic lithe aggregate bundle by by means means of of aa Möbius' transformation; all all those circles bundle Mobius' transformation; circles of M which pass pass through through the point oo form which form aa pencil pencil of straight lines, lines, interintersectingat at aa point point 00 of the plane. secting plane. Let Let A, A, B B be be any any two two straight straight lines lines through 0, 1L Since, 0, and and let letC0 be be any third third circle circle of M. Since, by by §§ 19, the the circles circles orthogonal circle, circle,the the point point 00 must be A, B, C0 cannot cannot have have a common common orthogonal be 0, and andconsequently, consequently, by by§ circles A, B, C0 belong belong to interior to C, § 16, the circles elliptic bundle, bundle, the the circles circles of of which which can all be be obtained obtained from from A, B, B, C0 an elliptic constructionofofpencils. pencils. And Andthis this bundle bundle must be by successive successive construction be idenidentical with with H, M,since since2W J1 is same construction. construction. is obtained by precisely the same
21. This § 16, shows showsthat that any three coplanar Thislast lastresult, result,together togetherwith with§ circles which do donot not belong belong to to the same same pencil pencil determine determine exactly exactiy one one circles which bundle.
22. AAbundle bundleofofcircles circles cannot cannot contain contain an an elliptic elliptic pencil pencil together together with with pencil. For 17, neither neither an elliptic its conjugate hyperbolic hyperbolic pencil. For since, since, by § 17, elliptic bundle can can contain contain an elliptic nor a parabolic parabolic bundle elliptic pencil, pencil, the given given bundle would necessarily necessarilybebekyperbolic, hyperbolic,sosothat thatthere there would wouldbe be aa circle, would circle, the orthogonal circle of of the the bundle, cutting all members orthogonal circle bundle, cutting members of the two two givell given pencils orthogonally; orthogonally; but conjugate pencils but this this isis impossible. impossible.
23. 23. Inversion Inversion with respect respect to a circle. circle. Givenaa straight straightline lineaaand andaapoint pointP, F, let let P* P bebethe Given theimagepoint imagepoint of of P in a; P the to a.a. More Pin a; we we shall shall call P* theinverse inverse point of P with respect respect to More generally, fixed circle circle A, we we can, Mobius' transformation, generally, given given a fixed can, by a Möbius' transform A into a straight for every every point point PP there transform A straight line; line; consequently consequently for is a point P inverse and P* P isischaracterised point P* inversetotoPPwith withrespect respect to A, and characterised by the the fact that that every circle through through PP and orthogonal A also passes by every circle orthogonal to A passe:> through through P*. The operation operation of of inversion inversion is involutory; further, further, the thefigure figure formed formetl by a.a circle A A and and two two inverse inverse points is is transformed transformed by by any anyMöbitis' Mobius' transformation into aa circle A and transformation into circle A and two two inverse inverse points. points. Thus, since' sine~·
12 12
MOBIUS TRANSFORMATION
[CHAP. [CHAP. I
inversion inversion with with respect respecttoto aa straight straight line gives gives an au isogonal isogonal but but not aa conformal representation plane on on itself, itself, inversion inversion with respect conformal representation of the plane A does does so so also. also. circle A to a circle
If tt isis an arbitrary 24. If arbitrary point point of of the the complex complex tt plane, plane, its itsinverse inverse with with axis is is given given by by the the conjugate conjugate complex complex number number t; t; respect to respect to the real axis more generally, generally, the points 19 t, x=e y=e19 i ...... (24:1) z=e1°t, are inverse inverse points points with with respect respect to to the the straight straight line through the origin are The first first of of obtained obtainedby byrotating rotating the the real real axis axis through through an an angle angle 0.8. The equations (24•1) gives x=eili, so so that
y=~x.
. ..... (24•2) (242)
Similarly, since Similarly, since the the equation =a a(l (1++it) it) X=
lit
...... (24•3)
transforms the the tplane tplane into into the the circle circle IxI= follows transforms the real real axis of the = a, it follows that the points 
a(1+it) a(l+it) a(l+ii) a(1+it) ...... (24•4) (244) 11—it it ' y = I1—it  H are inverse points with respect from this we respect to to 1xI = a; from we at once deduce that ...... (24•5) X=
25. Two Twosuccessive successive inversions inversions are equivalent either to a Mobius' Mobius' transformation or or to to the identical If, for for example, identical transformation. transformation. If, example, the information versionsare are performed performedwith withrespect respecttoto the the straight lines lines through the versions origin B+cp with with the the real realaxis, axis,we we have, have, by by (24·2), origin Owhich 0 which make angles 0, 0,8+4, and consequently ...... (25•1)
The resulting resulting transformation transformation isis therefore thereforeaa rotation rotation of the plane about 24,; thus thus the the angle angle of of rotation rotationdepends depends only the origin through an angle 2cp; on on the angle angle between between the the two two given given straight lines, not on the position qf these lines. Similarly, Similarly, for for two two inversions inversions with with respect respect to the the concentric concentric circles circles l.xxI=a I=a and and IxI=b, xl =b, we have, 1w hv (24·5), and so
a2/2, = a•;:c, y=
zz = b2/fj, ...... (25•2) (252)
The resulting resulting transformation transformationisistherefore thereforea amagnification magnification(Ahnlich/ceits. (Aknlickkeits
§§ 2427]
GEOMETRY OF THE TRANSFORMATIONS GEOMETRY OF
13 13
tranq'O'f'mation) transformation)which whichdepends dependsonly onlyononthetheratio ratiob b:aaof of the the radii, radii, not not on of the the radii radii themselves. themselves. on the lengths of Similarly seen that two two successive successive inversions inversions with Similarly itit is is seen with respect respect to parallel straight lines lines are are equivalent equivalent to to aatranslation, translation,which whichdepends depends parallel only on the distance distance between between them, them, on the the direction of the parallel lines and the but not not on on their their position position in the the plane. plane. two circles can, by means means of a suitable 13, any two Since, by by §§13, circles can, suitable Mobius' Mobius' transformation, of the the above above three threefigures, figures, we we transformation, be be transformed transformed into one of have proved the the following following theorem: theorem: two twosuccessive successit·e inversions of the the plane plane with of circles circles A, B are are equivalent equivalent to with re..~pect respecttotoan an arbitrary arbitrary pai1· pair of to a Jiu"bius' transfO'l'mation;the thesame sametransformation tranq'01·mationisisobtained obtainedbysuccessive by successive Möbius' transformation; inversions with with respect respect to to two two other suitable circles inversions circles A1, A 1 , B1 B 1 of the the pencil defined by A, A, B; may be be taken taken arbiB; and andone one qf qf the the two two circles circles A1, A 1 , B1 B 1 may defined by Further: the trarily in in this tkis pencil. pencil. Further: theresulting resulting.TJfôbius' IYiubius' tran!i/O'f'mation leavesall all circles circlesqf qi tke thepencil pencilconjugate conjugatetotoA, A, BB invatiant. invariant. lea1•es
26. Geometry of Möbius Mobius Transformations. 26. Geometry Transformations. planeexist existwhich whichare areinvariant invariant for for the the transtransPoints of the complex complex plane formation ...... (26'1)
x; i.e. For these points yy = =.x; i.e. they are the roots roots of of the equation equation y:lf + (8a.)x {3 = 0. . ..... (26·2) If all the thecoefficients coefficients in this equation equation vanish, vanish, the the given given transformation transformation If all is the identical identical one one py = x, and every point of the plane is a fixed fixed point. be the the roots of (26'2), so If x 2 be so that 0, let .x1 , x2 If y *. 0, X;=
a— S± a.8±JD 2 '
'Y
) ...... (263) (26'3)
D =(a.+ 8)2 4 (a.8{3y);
=00 or D =F 0. thus the according as as D D= thus thenumber numberof offixed fixed points points is is one or two according If = 0, the point point x = IX) is to be be regarded regarded as as aa fixed fixed point, so so that Ifyy =0, in this fixed points points is one this case case also the number number of fixed one or two according according as D=O D = 0 or D =F 0. then a = 8, and (26 0, thena.=8, 1) is of If also 27. Suppose Suppose first D=O. also y=O, (26"1) first that D = 0. If the form {3 (271) ...... (27"1) y=:r:+. (1. This is a translation 'fhis translationand andcan canbe beobtained obtainedby bytwo two inversions inversions with with respect respect
14
(CHAP. [cRAP. II
MOBIUS MOBIUS TRANSFORMATION TRANSFORMATION
If however to parallel straight straight lines. lines. If however y
** 0, 0, (263) (26·3) gives gives
ot8 .X1 = .X2 = 2y •
(272) ...... (27'2)
But on solving solving the the equation 1
1
2y
= ·+~ ...... (27'3) (273) y.x y—x11 x.x1 ot+o for for y, y, we we obtain obtain aa Mobius' Mobius'transformation transformationidentical identicalwith with{26'1)*. (261). If If we now introduce new coordinates now w=l/(y—xi), w=1/(yx1), t=1/(xx1 ), translation, namely the transformation again becomes becomes aa translation, namely w=t+2.L. ot+8
hrmat itrn We therefore have the the following following theorem: theorem:any anyMöbius' Mijbius'transj tranif!01'mation (26i)for which is zero th.ediscriminant discriminantDDis zerocan canbebeobtained obtainedby bytwo twosucsuc(26'1)/01' whickthe cessiveinversions inversionswith with t·espect respect to to two two circles ·whick which touch touch each each othet. other. cessive
*
*
secondlythat thatDD * 0.0. If7 28. Suppose secondly If y = = 0, then ot * 6; 8; and and by by putting
p
$
t=.xoot'
we we obtain, obtain, on on elimination of .x x and y, y, ot
...... (28'1) (281)
w=a t.
*
If on the other hand yy * 0, 0, write write y .:r. x.x. W=, t=    · y—z1 y.x~
x.x~'
...... (28'2) (282)
coordinatesthe the points points tt ==0, fixed points points of with the new new coordinates 0, tt = oo must be fixed the transformation, be of of the the form form transformation, which which must must therefore therefore be w—pt. w=pt. . ..... (28'3) (283) cn we we have have y = otfy, t =1; Correspondingtoto the the point zx == oo Corresponding = 1; thus thus (283) (28'3) shows that that shows = = ot y.x2 p=w=. ot— y.xl
equation may also also be be written This equation ot+8+.JD  ot+8.JD.
p·
...... (28'4)
•* This This is seen. hom transformations transformations transform the the seen. for for e:mmple, example, by bynoting noting that that both points z' = 10 x"= z" = oo, x" = — lif"'f ö/'y into into the points y'=x y' = x1, x'=x :r!"=y"'= e. y" = ely, y'" 1 , y"=a.f"'f, = oo.
§§ 2830]
GEOMETRY GEOMETRY OF THE THETRANSFORMATIONS TRANSFORMATIONS
15
Since, reduce to (28"1), the the case case Since,ifify=O, y=0, equations equations(28·3) (283) and and (28"4) (284) reduce to (281), when when 1'7=0 = 0 need need no no longer longer be be treated treated separately. separately. 29. If transformation ww = = pt is a magnification if p is real and positive, the transformation can be be obtained by by two two inversions inversions with respect respect to to concentric concentric circles. circles. and can is a rotation of Secondly, if Ii ppI==1,1, i.e. if if pp == e16, the transformation transformation is Secondly, if and can can be obtained obtained by by two two inversions inversions with with respect respect to to two the plane, plane, and intersecting straight lines. lines. If neither neither of of these these conditions conditions is is satisfied, satisfied, then then pp = cu/11, where 06 $ 0 (mod 2w), 2ir), a> a 0, and and aa'* * 1. (mod 1. The The transformation transformation can be be obtained obtained by aa rotation followed followed by a magnification, magnification, i.e. by four four successive successive inversions inversions Since, as as is is easily easily seen, seen, no no circle circle is is transformed with respect to to circles. circles. Since, not possible possible to to obtain obtain the the transformation transformation by two two inversions inversions into itself, it is not only. The only. Thetransformation transformationisisininthis thiscase casesaid saidtotobebelo.xodromic. lo:codromic. The various various cases cases can can be be clearly distinguished' by introducing introducing the 30. 'rhe distinguished'by parameter A (oc + o)2 44 (oco— {1y)"
...... (30'1)
Equation (28"4) now takes the form form (284) now JA + J(A1) ,J(A —1) JA+ p=JA.j(A1)'
and on on solving solving this this for for A A we obtain A= (p + I_r = 1 + 5_p_ 1)2 . —
4p 4p
—
4p
If now pp is real, real, positive positive and different from from unity, unity, A isis real real and and greater greater If 16 (6 than unity; (0 $ 0 mod = cos2 uuity; secondly, secondly, ifif pp = ee'° mod 2w), 211"), AA= cos2 1 0, 6, i.e. A A is is real, positive and and less positive less than unity; unity; finally finally AA = 1 if 1) D = 0. Thus : the tke transtrans0. Thus: formation if A is is not real or if real and formationisisalways alwaysloa,odrornic lo:codromic ij ifA A is real andnegative.. negative. If IfAA is is real rl'..al and andpositive, po:;ititoe, the tke transformation tran.iformationcan canbebeobtained obtainedby by inversions inversions with respect respect to to two witk two circles circles of an an eizptic, elliptic,parabolic, parabolic,or orhyperbolic hyperbolic pencil accordingasnsA>1, A=1, orA 1, A= 1, or A< 1.
CHAPTEB II CHAPTER
NONEUCLIDEAN GEOMETRY GEOMETRY 31. 31. Inversion Inversion with respect respect to the circles of aa bundle. We shall transformations which shall now now consider consider the theaggregate aggregate of ofMUbius' Mobius' transformations which are obtained by two successive inversions two successive inversionswith withrespect respecttoto circles circles of of aa given bundle. It will first be be proved proved that these transformations form form a It will group. In the the first first place, place, the thetransformation transformation inverse inurse to any one of the transtransformations considered the two two formations consideredisis obtained obtainedby by inverting invertingthe the order order of the inversions; for four inversions withrespect respecttotothe the circles circles A, A, B, B, B, A, inversions; inversions with taken in in the the order order indicated, indicated,clearly clearly produce the taken the identical identical transformation. transformation.
form aa group, group, it re32. To Toprove provethat thatthe thetransformations transformations considered considered form mains to thatfour foursuccessive successive inversions inversions with respect to the the circles circles to prove prove that A, B, C, C, D of ofthe thebundle bundlecan canbe bereplaced replaced by by two t.wo inversions with respect to circles circles of the same bundle. in two two Suppose in Suppose in the first place that the two circles circlesA, A, BB intersect in pencil. It Q, thus defining a hyperbolic hyperbolic pencil. It follows follows from § 22 that points P, F, Q, points of of the the pencil pencil determined determined by by C P and and Q Q cannot both be limiting limiting points thereisis at at least least one one circle circle C, C, of of this pencil and D, and consequently consequently there C, passes passes through through P. P. Q; suppose C, points P, passing through one of the points F, Q; By§ B, C, C, belong belong to the same same pencil, and consequently By § 18 the circles A, B, to A, B can (§25) the two inversions inversions with with respect respect to can be be replaced replaced by by ininversions withrespect respecttoto A,, A,, C,, 0" where where A, A, is is a circle circle of this this pencil. pencil. versions with Similarly, the replaced by inversions inversions to C, 0, D can be replaced Similarly, the inversions inversionswith withrespect respect to respect to to C, C,, D,, D~> where where D, D, is is aa. circle circle of of the pencil pencil determined by with respect C, to A, A, B, C, are therefore 0, D. The Thefour fourinversions inversions with with respect respect to C, B Dare therefore equivalent to four four inversions inversions with withrespect respecttotoA,, A,, C,, C,, 0,, C,, D1, D" i.e. i.e. to two inversions with respect respect to to A,, A,, B,, D,, since since the thetwo twosuccessive successive inversions inversions inversions with with respect to C, 0 1 destroy destroy one one another. another. determinedby byB,B,CCcontains contains at at least In the the general general case the pencil pencil determined either coincides with AA or or intersects intersects A A at two points. coincides with points. one circle B, which either B,CCmay maynow now be be replaced replaced by by inversions inversions The inversions with with respect respecttotoB, with respect to B, B,, C,, 0 11 and andconsequently consequently the inversions inversions with with respect to A, B, C, B by by inversions inversionswith withrespect respecttotoA,A,B" B,,C,, C,,D; B; then either the C, D the oneanother anotheror or the the problem problem has inversions with respect to A, B, destroy destroy one been reduced reduced to the case been case already already dealt dealt with. with.
§§ 3135] 31—35]
GROUP OF TRMISFORMATIONS TRANSFORMATIONS
17
just been The groups of transformations, transformations, the existence of which which has just been established, are according as bundle conconestablished, are fundamentally fundamentally distinct distinct according as the bundle sidered is elliptic, parabolic, parabolic, or or hyperbolic. hyperbolic. 33. The Thecircles circlesof ofan an elliptic elliptic bundle bundle can can be be obtained obtainedby bystereographic stereographic great circles circles of a suitable sphere. Any Anyinversion inversion of of the projection of the great circle of the bundle bundle corresponds corresponds to an ordinary ordinary plane with respect to a circle of the thecorresponding corresponding inversion of sphere with inversion of the sphere with respect respect to to the the plane of great circle, circle, and the the group group of ofMöbins' Mobius' transformations transformations obtained obtained is is isomorphic the sphere. sphere. isomorphic with with the group of rotations of the
34. The Thecircles circlesof ofaa parabolic parabolic bundle can be transformed transformed 'hya. suitable bya suitable Möbius' transformation transformation into into the aggregate of straight the plane. plane. Mobius' straight lines lines in in the Since every ordinary motion of the plane plane can can be beobtained obtainedby bysuccessive successive two straight straight lines, the group of transformations inversions with respect to two isomorphic with with the the group group of motions motions of of aa rigid plane. now isomorphic is now
35. Representation 35. Representation of aa circular area area on on itself. itself. The most most important important case casefor forus usisisthat that in in which which the the bundle bundle is is kyperThe bolic, so that that the bolic, so the group group considered considered is obtained by inversions with respect to two straight line) two circles circles which which cut a given circle (or straight line) orthogonally. orthogonally. It is seen immediately It immediately that that for for each each single single inversion, and therefore therefore for for every operation of the group, group, the the circumference circumference of of the the orthogonal orthogonalcircle circle is transformed into itself, and the interior * interior of of this this circle circle into into itself itself•. transformed into itself, and now prove that, that, conversely, every8IObius' Zktöbius' transformatisnqf ofwltick We will now conversely, every transformation It is convenient to prove tkis is true is a tran.iformation of tlte group. It is convenient to prove this true is a transformation of the group.
that the this theorem theorem step step by by step; step;we we shall shall suppose suppose that thecircular circulararea area.which which transformedinto intoitself itselfisisthe theunitcircle unitcircleIz I ~ 1. is transfermed that any We first prove prove that any interior point Q Q of this area, with coordinate can transformed z == a (I £t I < 1 ), can be transformed j 1), a (I traiisorigin F, P, zz = 0, by a transinto the origin formation of the group. group. The The point point respect to the Q Q11 inverse to Q with respect to the unitcircle 0 has unitcircle has coordinate coordinate 1/a, 1/ii, and Ia and and the thecircle circleCCwith with centre centre11/cl
radius
J c~ 1)/ —
is orthogonal is orthogonal
to 0. Now Nowinvert invertfirst first with with respect to the circle circle C and then then with with rerestraight line line PQ, PQ, both spect to the straight
Fig. 44 Fig.
is a.a straight straight line, line, ea.ch each opera.tion operation of the group •* If orthogonaJ circle is group transforms If the orthogonal each of the halfplanes defined bythis this stra.ight straight line into itself. ea.ch defined by itseU.
18
NONEUCLIDEAN GEOMETRY GEOMETRY NONEIJCLIDEAN
[ca&p. ii (CHAP. II
of which of the thehyperbolic hyperbolicbundle bundleconsidered. considered.The Thefirst firstinverinverwhich are circles of sion transforms Q into into F, P, and andthe thesecond secondleaves leavesFPfixed, fixed,sosothat thatthe thefinal final transforms Q result is to to transform transform Q Q into P. The following observation will will be of following observation of use use later: later the : theabove aboveMöbius' Mobius' transformation transforms transforms·the the points transformation points a, a/..J(aa), af J(aa), 1/a into the points points and is 0, — a/ J(aa), oo respectively, and is therefore therefore (§ 11) given given by by the equation a:r. (351) y = 1 ax. ...... (35"1) Since rotations about about the theorigin originbelong belongtotothe thegroup groupconsidered, considered, the the that has result that has just justbeen been proved proved may be stated in th: :bll4wing :oll~wing sharper form: we area Izz I t;; 11 form: we can, by an operation operation of the the group, group, transform the area into itself itselfin insuch suchaaway waythat thata adirected directedlineelement lineelementthrough throughan aninterior interior point of the directed point the area areaisistransformed transformedinto intoa given a given directedlineelement lineelement through the origin. origin. In In particular particularthe thetransformation transformation
xa
'!! = 1~·&:X·
(352) ...... (35"2)
being obtained obtainedfrom from(3.51) ( 35 ·1)by byrotation rotationthrough through180', 180",belongs belongstotothe thegroup. group. being that a Mobius' The complete complete theorem theoremwill willhave havebeen beenproved provedififwe weshow showthataMobius' which represents the the circle circle on on itself itself in in such such a way w11.y that transformation which a given lineelement through through the theorigin origin(and (andconsequently consequentlyevery everylinelineelement through through the theorigin) origin)isistransformed transformedinto intoitself itselfmust mustofofnecessity necessity be the identical identical transformation. transformation.But, But,for forsuch suchaatransformation, transformation,the thepoints points z = 0, :c points, and the point :c ==11 isis transformed z == oo are fixed :c fixed points, transformed into into a 9• The transformation = e" :r., transformationmust musttherefore thereforebebeofofthe theformy form !I= point z:c ==ei e'°. from the invariance of and it it follows follows from of lineelements lineelements through through the the origin origin that 0 0. that()=
NonEuclidean Geometry (9). 36. NonEuclidean Geometry (8)(8)(9). 'fhe The group group of ofMöbius' Mobius' transformations transformationswhich whichrepresent representaacircular circular area (or a halfplane) on itself itselfhas hasmany manyproperties propertiesanalogous analogoustotothose those of of the group of of motions motions of a rigid plane. group plane. In In this thiscomparison comparison certain certain circular circular the circular circulararea area take takethe theplace place of of straight straightlines lines inin the theplane; plane; arcs in the these circular circular arcs arcs are areininfact factthe theportions portionsofofthe thecircles circlesofofthe thehyperbolic hyperbolic bundle which which are are inside the circular bundle circular area area or or halfplane. halfplane. Thus, corresponding correspondingtotothe the fact fact that that aa straight straight line line in in the the 'l'hus, Euclidean plane is determined Euclidean plane determined uniquely uniquelyby bytwo two points pointson onit,it,we we have have the immediate theorem that thatthrough throughany anytwo twopoints pointsininthe theupper upperhalfhalfplane (or inside the zI= the circle circle II:c andonly onlyone onecircle circlecan canbe bedrawn drawn = 1) one and to cut cut the the real real axis axis(or (orthe theunitcircle) unitcircle)orthogonally. orthogonally. Again, Again,ititfollows follows from what what has already from already been said that thatone oneand andonly onlyone onecircle circle of of the the
§§ 3638]
ANGLE AND DISTANCE DISTANCE
19 [9
hyperbolic hyperbolic bundle bundle can can be be drawn drawn through through any any given given lineelement lineelement in in the upper halfplane (or in the circle circle IxxI < 1); 1); a precisely precisely analogous analogous statefor straight lines in the the Euclidean Euclidean plane. ment holds for In virtue virtue of of this thisanalogy, analogy, the the circular circular arcs arcs in in question question will will be be called called nonEuclia,ean tke nonnonnonEuclideanstraight straight lines, lines,the the halfplane halfplane (or (or circular circular area) the transformations which which transform transform the the Euclidean plane, and the the MUbius' :Mobius' transformations nonEuclidean be called called nonEuclidean nonEuclidean motions. motions. nonEuclidean plane into itself itself will will be 37. One Onefundamental fundamentalcontrast contrastwith withordinary ordinarygeometry geometryisishowever however seen seen at once. once. For For in inEuclidean Euclideangeometry, geometry, by by Euclid's Euclid's 11th postulate, through which does line one one and and only only one one any point which does not iiot lie lie on on aa given given straight line straight line can be drawn whichdoes doesnot not intersect intersect the given drawn which given straight line; whereas, P which which whereas, in the nonEuclidean nonEuclidean plane, plane, through any point P nonEuclidean straight does does not not lie lie on on a nonEuclidean straight line oc an infinite number of nonEuclidean nonEuclidean straightlines linescan canbe bedrawn drawnnone noneof ofwhich which straight two nonnonintersect oc. Further, there are two Euclidean straight lines f3 and yy through Euclidean through which divide divide all all the the remaining remaining nonnonP which Euclidean straight straight lines lines through through PP into Euclidean two classes: namely those which which intersect intersect and those do not. oc and those which which do not. Two Two nonnonEuclid canstraight straight lines such as oc and {3, Euclidean which are in in fact fact two two circles circles which touch Fig at aa point point of ofthe theorthogonal orthogonal circle, circle, are Fig. 55 called parallel parallel (Lobatschewsk!f, 11793—1856). called i931856). 38. 38. Angle Angle_and distance. NonEuclidean motions, being being Mobius' Möbius' transformations transformations which which interNonEuclidean motions, change change the nonEuclidean nonEuclidean straight hues, lines, leave unaltered the the angle angle between between two intersecting straight straight lines, lines, and and consequently consequently the ordinary taken as the nonordinar!Jangle angle can can also also be taken nonEuclidean angle. angle. ordinary distance The ordinary distancebetween betweentwo two points, on the other hand, is not invariant invariant for for nonnonEuclidean motions, motions, and and an invariant invariant funcfunction of two two points must be determined determined to to replace idea of of distance distance is is to to be be replace itit if the idea Fig. 66 employed Fig. employed in in nonEuclidean nonEuclidean geometry. geometry. Let P, Q be two points nonEuclidean plane, and let S, TT denote denote the the two points of a nonEuclidean plane, and let 8, F, Q extremities of of the the nonEuclidean straight line line joining joining PP and Q. Q. Then extremities nonEuclidean straight I
J
20
[ClAP. II (CllAP.
NONEUCLIDEAN GEOMETRY NONEUCLIDEAN
the crossratio crossratio A ,\of P, Q, of the the four points (S, F, Q, T) is uniquely determined by the two two points P, Q, and this crossratio crossratio is invariant for for all all nonnonF, Q, function tfr (A) (.A) of of,\A isis also We Euclidean motions; hence any function alsoinvariant. invariant. We (A) so shall now so that, that, if it is D (P, (F, Q), now choose the function function i/'tfr (.A) is denoted denoted by D Q), and if R B is is any any point pointwhatever whatever of of the the segment segment PQ PQ of of the the nonEuclidean nonEuclidean line through through P F and straight line and Q, Q, then D (P, D D(P, (P, R) R) + +D D (R, Q) = D Q). . ..... (38"1) (B, Q) (F, Q).
.numbers whose whosesum sumisis less less than positive .numbers 39. Let Ith and ~It be two positive unity; denote denote by by 00 the thecentre centreof ofthe theorthogonal orthogonal circle circle I xII = 1, and by P, Q Q the points h, It, hIt+ (38"1) gives gives + tl.k. Then (381) F, D(O, Q)=D(O,P) . ..... (39"1) = D(0, F) +D(P, + D(P, Q). (391) D(0, Q) Q). Now the crossratio of the — 1, 0, 0, It, k, 11 is is a function Now the four four points pointsfunction of of ii, h, so that we may write that we D(0,P)—4)(h), D (0, P)= cp (h), D (0, Q) = cp (h + tl.k). . ..... (39"2) (392) apply the To calculate calculate D (P, Q) Q) we apply To D (F, I
nonEuclidean motion (§ 35)
xk
y=1kx; Y = this transforms transforms P and Q Qinto the P into 00 and point + tl.k)}, ii (h — It tl.k/{1(It+ and consequently, since distance is to be invariant, D(P, Q)= N that from the follows that the point pointof ofview view of nonL1 Euclidean limitrotations Euclidean geometry geometry all limitrotations are equivalent, or rather that equivalent, or that they they can can differ only sense of the rotation. differ only in in the sense For the limitrotation by For limitrotation obtained obtained by inversions with respect respect to to LM LM and and L.!I.T LV inversions with M Fig. 11 is the theonly onlyfixed fixed point. Circles Circles inside inside Fig. 11 L is 1
§§ 46—48] 4648]
PARALLEL CURVES
25
the nonEuclidean nonEuclidean plane plane which whichtouch touch the the orthogonal orthogonal circle circleatat LL are are transformed into themselves; these thesecircles circles are arecalled calledoricycles. oricycles.
47. Every Everyordinary ordinarycircle circle which which lies lies wholly wholly within the the nonEuclidean nonEuclidean nonEuclidean circle, find its nonnonplane plane isis also also aa nonEuclidean circle,and anditit isis easy easy to to find Euclidean Euclidean centre. centre. Similarly every every circle circle which which touches touches the the orthogonal circle is an oricycle, and every every circular arc whose endpoints lie lie on on the oricycle, and whose endpoints the circle orthogonal orthogonal circle circle is is a hypercycle. hypercycle. The curves curves of these these three three types types are only curves curves of of the the nonEuclidean nonEuclidean plane which which have a constant constant nonnonthe only zero By §§ 44 the curvaturecurvature· of of the theoricycle oricycleisis(disregarding (disregarding zero curvature. curvature. By 2, while that that of ofaa hypercyele hypercycle is is less less than than2,2,and andof ofaa nonnonsign) equal to 2, 2. If k denotes denotes the Euclidean circle circle greater greater than 2. the curvature curvatureand and rr the circles, we nonEuclidean nonEuclidean radius radius of of one of these circles, we obtain obtain the the relation relation e"" +e_,. k=2 e"" t:r=2coth2r.* ...... (47·1) e
48. Parallel curves. 48. curves. Consider the the aggregate Consider aggregate of nonEuclidean nonEuclidean circles circles with with given given nonnonEuclidean radius r whose whose centres centres are at the the points points of an arbitrary set A of points of the nonEuclidean nonEuclidean plane. These These circles circles cover cover a set of points B (r), whose frontier, frontier, if if itit exists, exists, contains contains all all points points of the nonEuclidean nonEuclidean from A. plane which which are are at at a distance r from If we take as as the set A aa curve and if we we let let rr If we take curve C 0 of of finite curvature, curvature, and vary while while remaining remaining less less than aasufficiently sufficiently small small upper ~pper bound, bound, we we obtain obtain a family family of parallel parallel curves curves (in (in the sense sense of the the JlOnEuclidean metric). It can metric). Ca.n be be proved proved in in just justthe thesame sameway way as as with with other other similar similar theorthogonal orthogonal trajectories trajectories problems problems of of the the calculus calculus of ofvariationst variationst that that the of such a family of equidistant curves curves are are nonEuclidean nonEuclidean straight straight lines. lines. of the thenonEuclidean nonEuclidean plane planeisissimply simply covered covered Conversely, Conversely,ifif aa portion of by a family nonEuclidean straight straight lines, the orthogonal family of nonEuclidean orthogonal trajectories trajectories of the above. Thus the the the family family are are parallel parallel curves in the sense sense defined defined above. families of parallel parallel curves are the simplest examples of families the following: following : common centre; centre; (a) nonEucliclean nonEuclidean circles circles with with aa common (b) oricycles oricycles which which touch touch the orthogonal circle circle at at aa common common point; point; (c) hypercycleshaving same endpoints endpoints M, M;,21(2 M 2 (see Fig. 10). hypercycles having the same 11"
r the wplane, wplane, i.e. writing zz = =a:+ replace ((491) 49·1) by in the z + iy and and w == pt!+, we replace two equations equations the two p=ex, p = e", çb=y. 4> = y. A horizontal strip of the iplane zplane bounded hounded by the lines yp = y, and '!J y,, y = P2, wedgeshaped region where I y,y transformed into a wedgeshaped region of of the Y22 I1< 2r, is transformed I
0
zplane plane
w— plane Fig. 12 Fig.12
wplane, the angle angle of the wplane, the the wedge wedge being being ex= =II y,— !It 1 The = Ilt/>2— t/J, II= representation is is conformal throughout the interior representation conformal throughout interior of of these these regions, regions, since neverzero. zero. since the derivative derivative of of?e"isisnever As a special special case, case, if IJ/2 — r (e.g., y1 y1 == 0, 0, 'P2 !J2 == ir), '~~"), the wedge wedge behe y,l = 7r comes a halfplane. strip, namely namely that II'Yi !J1  Y2! y2 1O. dw) = 2klogk (fdw\ 1—k2 dz •=O 1k —
2
. ..... (572) (57•2}
* Alliernau4 points of the * A Bieman.J surface is simple (schlicht) (•chlicht) if no two points ihe surfacehaveihesame surface have the same coordinate u.
§§5760] 5760]
31
EXTERIOR OF THE ELLIPSE
58. IfIfn nisismade madetotoincrease increaseindefinitely, indefinitely,the theRiemaun Riemann surface surface dealt dealt with with 58. Riemann surface surface of of§§57, having a becomes,ininthe the limit, limit, the Riemann 57, having 56 becomes, in §§56 logarithmic branchpoint. It It is logarithmic branchpoint. is therefore therefore to be be expected expected that, that,ifif4',. ~. (z) (z} denotes member of of (56"2), (562), and 4' (z)that that of (571), denotes the righthand righthand member 1/J (z) (57"1), we we shall shall haVe have ~.(z} . .•... (58"1) lim 4, (z) = 1/J(z). 'Ii (z).
..
,._
The truth truthof of(581) (58"1)can canininfact factbe bededuced deducedfrom from aa general general theorem; but verified directly, directly, as as follows. follows. In the equation equation can also be verified In 1
(Jii. )" =t= 11
put
I
I _! 6,. {log(ll.nz)lotrl1ll"•l}
••••••_(~8"2)
(1k"'z)"' urn e,.=0; lim exponent of (58"2) can can = 0; the exponent of e in (582)
1— i,.,soso that that k"= 1e,.,
.......
now be written as 1lo n{log(1ze..)log(1z+e,.z)} g og log(1e,.) · ' and when when e,. tends to zero zero this this tends tends to to the limit 1 +z 1+z log ii. logk. 1
i— z
Equation Equation (581) (58"1)now nowfollows follows at at once. once. It It can can be be shown shown in the the same same way that the the limiting limiting form form of (56"3) is (5V2). (57"2).
59. ItItcan canalso alsobebeshown shown that, that,for forall allvalues values of ofn, n, ~'.. (0) > ~·.+1 (0), ...... (59"1) and this inequality may perhapsrest restuponsorne upon somedeeper, deeper, as as yet unremarked andthisinequalitymayperhaps 1
— A,., so h"1 = property of the transformations. transformations. To To prove prove (59"1) write log li =~.so that .\,.+1 > r> r> 11 (or the interior of the circle circle I z I < ~ 0; by by adding adding to y, a portion portion of the axis u> u > 00 we we therefore therefore obtain obtain which lies lieswholly whollyininT,T1and andsurrounds surroundsthe thepoint point uu=0. = 0. a closed curve which But no nq such curve curve can can exist, exist, since since T1 T, isis simplyconnected simplyconnected and and does not contain either of the points uu == 0, 0, u = oo. contain domainwhose whosefrontier frontiercontains containsat at least Thus: any simplyconnected simplyconnected domain two distinct points can, tmn.iformations, be be conformally conformally repretwo can, by by simple simple transformations, sented m; a domain which which lies entirely inside the unitcircle. unitci1·cle. A multiplyconnected multiplyconnected domain domaincan can be be treated treated in A in precisely precisely the same same way provided provided that that its frontier frontier contains contains at at least leastone one continuum continuum of ofmore more way than than one one point. point. though in appearance little different different from from 65. AAtheorem theorem which, which, though appearance but little thatof § 64, that· of§ 64, is, is, on on account account of of the the value value of of the the constant constantinvolved, involved, of of the the greatest importance importance in the general general theory, theory, is is due due to to Koebel2). Koebe(l2}. * Each of of the the four four domains domains corresponding correspondingto to TT1, inparticular particular T7',, (schlicht) 1 , in 2 , is simple (•chlicht) (cf. §§56, 56, footnote).
36
ELEMENTARY TRANSFORMATIONS
III (CHAP. m
domain of the wplane not conLet T be be aa simple simple simplyconnected simplyconnected domain x;; suppose supposefurther furtherthat that a.ll all points points of the the point point w w= = oo the circle circle taining the T, but but that that w w==1 is a frontierpoint of T. T. Any I ww I l/(z0 ) I.I· I> 11(z0) nowaafunction/(z) function f(z) satisfying the following three conditions: conditions: Consider now following three (i) I1(z) (z) is is regular regular in the the circle circle Iz I
IIw(t)kitI. w (t) I ~ It I·
...... (10·2)
inequality can be expressed expressed by bythe thestatement statement that that the nonEuclidean This inequality (t) is is not greater distance .D (0, w (t)) of of the (0) = 0 and w distanceD w(t)) the two two points pointsww(O)=O w(t) than the the nonEuclidean nonEuclidean distance distance of the poil').ts 0, t, so that, remembering that, remembering that, as as was was proved in Chapter II, ii, nonEuclidean nonEuclidean distances distancesare are invariant invariant we obtain obtain the for the transformations transformations (70"1), we the following following theorem, theorem, which which includes includes Schwarz's Schwarz's Lemma Lemmaasasaa special specialcase caseand andwhich whichwas wasfirst first stated stated by G. G. Pi~kCl4l. PickU4. by THEOREM 3. 3. LetLet f (z) TaEOREM 1(z)bebeanananalytic analyticfunction functionwhich whichisis regular regular and denote any two such that lf(z) 11(z)I< 1—11(0)! >1—11(0)1 1lf 1I >1I/(O) I < 3) 72 . 1l z I ""1 + lt,.' (0) I< It/>2' (0) I· But, by (56·3), (0)1 1 +kp k>(1k)p, 1+r log hh I+r
I.e. i.e.
ll/t(u)kl>(1k)e 1 r
•.
...... (784) (78"4)
79. Consider Consider now now a function function w=/(z) having the three three properties properties (i), w =f(z) having (iii) of§ of §67, 67,and andhaving havingalso alsothe thefurther furtherproperty propertythat that there is a (ii), and (iii) h has no solution real number number k, h, (0 (00,
where x" is the circle IIzI= z I= 1. 1.
(816) .... 0.(81"6)
______
§ 82] 82)
KOEBE'S CONSTANT CONSTANT
49 4:9
On by the the Mean Mean Value Value Theorem, Theorem, On the other hand, by
r2· logpd8,
1 logicf»(O)I=I0ogcf»(0)= 27T}o
(811) ...... (81"7)
(0) = F'(O). F' (0 ). Comparison Comparison of(8P6) of ( 81 ·s)and and (811) (81"7)shows shows that and, and, further, further, cf»4(0)=
F'(O) log II F"(O)
I~ 2:.ilogpd.r. I
. ..... (8r8) (81"8)
82. Let 1(z) Let/ (z)be bethe thefunction function considered considered at the end of of§§ 80. 80. If If rr (t) II kI + T·
w
1
...... (83"7)
84. If a is an arbitrary arbitrarycomplex complex numnumber, represented (say) P, ber, represented (say)by by the the point point F, equal to to the the number aI a—  1I I, being being equal length of the segment segment UP, UP, is is not greater than the length length UHF, UMP, where where MP is is an arc of aa circle circle with withcentre centre0. 0. We We theretherefore have fore have the inequality 4 Iall IIa—i a  J I ~ JI IIaI a I  II + a II IEflog log aa.i . ...... (84i) (84"I)
Fig. 22 22
I
86. By (841), (84"1), 85. By —1 14> (t) 11 ~ 114> (t) III+ 14> (t) IIjf lE Iog4> (t) I· 141(t)—il
...... (851) (85"1) the branch %fr (t) =log = log 4> 41(t), But since, by (83"5}, 4(0) 4> (0} = 1, the branchofofthe thefunction function 1/1 (t) (t), satisfies, in in consequence consequenceofof(83"6), (836),the theinequality inequality which vanishes fort== for t = 0, satisfies, lllil/l (t) I~ 2log k; 76, if if It I ~ r, and therefore, therefore, by §§ 76,
(t)l ~  4 ~ogklog ~ ~~:
Also, by by (83"6}, (836), Also, 1 141(t)I 14> (t) I ~ Ji2'
114> II (t) 1 II~
...... (852) (85"2)
i—hk22 I
Ji2 .
. ..... (853) (85·3)
It from(85"I), (85i), (85"2) and and (85"3) (853) that It now now follows follows from l4> (t) _ II ~ I1—h2  k 2 _ 4logh 4 log It log 1+r I +r k2 7rk2 I r' (834) and and on using (83"4) and (831) (83"7)we we obtain obtain finally finally jw — z I ~m(k, m (ii, r), r), lwzl
(1—h3)'r 4r h3 ) r _ 4rlogh1 4rlop: k 1og .i!:_ (k ) =(Im 'h ', r) r — k" "Irk• og 1  r · —
l
•
0
••••
(85"4)
From this this e::pression expressionitit isis seen seen that that not From not only only isis urn lim m m (h, (k, r) ==0 0 for for h.1 h—i
every fixed fixed r, as was was to be be proved, proved, but but also also urn lim m m (h, (k, h) h)==0. 0. h1
BCHWARZ'S iwMMA SCHWARZ'S LEMMA
52
Iv [CHAP. IV
Bylonger longer calculations calculations than than the theabove above itit can can be be shown shown that Remark. By that function m m (h, (85·4) can can be be replaced replaced by aa much much smaller smaller the function (h,r)r) of of (854) function.
86. Problem. Problem. 86. wake use use of the results of The reader may now now make of§§ 82—85 8285 to establish the following following theorem. theorem. · the the tiplane Uplane which two simplyconnected domains of the Let Rv. and S,. be two in their their interiors interiors and and which which are are represented represented on on contain the point u ==00 in the interior interior of of the the unitcircle unitcircle IIz I > 0), respectively. respectively. It f' (0) (0) > 0), and u = = g (z), (g (0) == 0, g' (0) Itis is supposed 1' =1(z) and u = that the the functions functions u =f(z) = g (z) represent the circle circle Iz I O.
...... (903) (90"3)
Write now now u,. = 11 I v,. = 11 — If (z,.) I, and construct construct the circles circles Ks', K,.', — z,. I,, v,, Kand rF,,' ,.' of of §§ 89. Still using K and I'r to todenote denotethe theoricycles oricycles to to which which K,,' K,.' and FR converge, converge,let let Kn K,, and and r,. F,, denote denote circles circles with with the same nonEuclidean 1',.'
but having centres at z,, radii as K,,' Kn' and F,,', r,;, but having their their nonEuclidean nonEuclidean centres z,. and z,, and f(z,,) respectively, instead instead of of at at lznl I. Thus K,. f(z,.) respectively, 1/(z,.)l. obtained K,, is obtained
§§ 9092]
THEOREM JULIA'S THEOREM
55
from but since, since, by (90'1), from K,.' K,.' by by aa rotation rotation about about the the origin; origin; but (901), the angle infinity, the circles circles K,. K,. tend to of rotation rotation tends tends to to zero zeroas as nn tends tends to infinity, the same same limiting circles K,.'. Similarly the the circles circles F,. 1',. limitingfigure figureKK as as the circles tend to F. 1'. =1(z) cannot Thus, by by §§87, point zz lies lies inside inside K, the the point point w w=f(z) cannot 87, if the point lie outside 1'; we can of lie outside F; we can in in fact fact show, show,by byan anargument argumentsimilar similarto to that that of §§ 67, thatf(z) 1'. that 1(z) must lie inside I'. The above result constitutes Julia's Julia'sTheorem Theorem• * (16).
91. The work work of §§ 90 may may be be completed completedby byadding addingthat that ifif there there is aa point z1 of the frontier which is transformed of point z1 frontier of of K K which transformed into into aa peiatf(z point f(z,) 1) of 48, the frontier frontier of of F, 1', then then 1(z) f(z)must mustbe beaabilinear bilinearfunction. function. For, by §§48, touch the circle any two two oricycles oricycles which which touch circle Iz I== 11 at zz = 11 are parallel parallel curves thesense sense of ofnonEuclidean nonEuclidean geometry; geometry; also also equation equation (89'4), curves in the which nonEuclidean motion, motion, transforms the two two oricycles oricycles which represents a nonEuclidean into the the two oricycles oricycles through '!h through the the points pointsz1 .x1and and x2 .:v2 (v1 (.r1 0, the last .
I1/(x}l
!~11/(x)l
But, if if we we write
...... (93"5) (935)
I.
1f(x) = AeUI,
...... (936) (93"6)
where we have where A> A> 00 and and 06 is real, we 20080—A I1/(x)l_ 2cos6A . 1—If(x)I 11=/(X}T 1 + J(1 2A. cos 8 + A.2) ,
. ..... ((93"7)
also, when xx tends tends to unity, by (93"5) (935) also, when unity, A A tends to to zero, zero, and therefore, therefore, by must tend tend to to unity. unity. From this it and (931), (93"7), cos cos 06 (and (and hence hence also eUI) must From this follows, onusing using(93"5) (935) a.ud and (93"6), (936), that follows, on 1 —f(z) —f(x)I 1 f(x)leUI lim =at. . ..... (938) (93"8) = urn 1Ii urn 1 f(x)=lim 1—z 1—x 1x • 1x We may what has has been proved in in the form may collect collect what been proved form of of the thefollowing following theorem: .,_1
THEOREM. regularand and suck suck tkat that 11(z) I > N N,2 the oscillation that for oscillation 0,.11:1 < tE; finally, suppose suppose that for n> n >.Y3 N 3 the the point pointZn z,. denotes the greatest of the numbers numbers N1, N1, N2, N,, Na, N3, lies in ()1.1:1. Thus if N denotes we have, >Nand m > N, in addition addition to (966), (96'6), we have,provided providedonly onlythat thatnn> N and m> inequalities the inequalities lf,.(z,.)f I< !E, lf,.(z,..)f ... I< 1€....... (967) (96'7) if,, (z,,) .. Ifm (Zm) fm In view (964) to (96'6), view of (96'4) .. .... (96'8) I w,.w,.l <E (n,m>N). Hence, by by Cauchy's Cauchy's criterion, criterion, the sequence sequence w,, w,. converges, converges, so Hence, so that Theorem 3 is proved. proved. THEOREM 4. Q B denotes denotes tke whick the tke sequence sequence the set set of ofpoints pointszzof ofAA at which THEOREM (z) converges convergestotoa afunction f(z), then thenevery everylimiting limitingpoint pointZ;0 of f,. (z) function f(z), of B such suck that tkat w(A)(z = 0 belongs belongs to tkefunctionf(z), whick is is defined defined in to B, B, and the functionf(z), which in B, (z,) 0) =0 is continuous at z,. continuous at z0 • From 'l'heorem the functions functions In f,. (z) (z) From Theorem 33 we wehave haveat at once oncenot not only only t.hat that the converge at theconvergence convergence converge at zz0, sothat that zz0 isaa point point of of B, B, but but also also that that the 0 , so 0 is at iscontinuous. continuous. Now consider an arbitrary z,, ... at z0 z0 is arbitrarysequence sequence of of points points z1, z~> Zt, of B, having as limit; to 1(z) of having z0 z 0 as to establish establish the thecontinuity continuity of off (z) at at z,zowe we have only only to to prove prove that ...... {96'9) (961) limf(z,.) =f(zo)·
m m
I
m
m I
I
if
nco
Since the sequence sequence/,. each of of the the points pointsZk, z~;, we can find I,, (z) converges converges at each .an increasing increasing sequence ... , such that for for every k sequenceofofintegers~. integers n1,n,,n,,..., 1 lf,.t (z~:)f(z~:) I < k. I
(9610) .. .... (96'10)
Equation {96'9) now follows follows on fact that, that, Equation (961) now on combining combining(96'10) (9610) with with the the fhct on account of the continuous convergence of the the sequence sequence at at z,, convergence of zo. (9611) lim f,.t(z~;;)=f(zo)· ...... (96'11) urn fnk(zk)=f(zo). kco
§§97, 97, 98]
61
NORMAL NORMAL FAMILIES
97. Normalfkmilies ftunilleaof ofbounded functions(17) (17) 97. Normal bounded fUnctioni
(18).
We consider definite class {/ (z )} offunctions consider a definite which are defined of functions f(z), 1(z), which
in a domain R and and are areuniformly uniformly bounded bounded in incertain certainneighbourhoods neighbourhoods of every point such aa family family of offunctions functions we we can can define define for point zz0 ofR. B. For such 0 of each point we shall shall call call the the limiting limiting oscillapointz0 z0 of RB a number ow (z0), (z0), which we tion of the family. We Weagain againconsider considerthe thesequence sequenceof ofcircles circles 0m for all points points on on the circle circle II zz — z0 I = p, provided that nn isissufficiently sufficiently circle/,. (z) =1= if n is is sufficiently sufficiently large, large. large. But But in in this this circle!,, large. * 0, and therefore, if (1032) If,. (Z0 ) I~ m. . ..... (103"2) Thus lf(zo)l ==lim lim If,,(zo)I lf.(zo)l ~m, ..._
..
that and it it follows follows t~tf(z * 0. 0)=1=0. this theorem theorem is is:: An immediate corollary corollary of this THEOREM dinnain B functions!1 (z),f2 (z), (z),... THEOREM 4. 4. IfIfininaa domain R aa sequence sequence of functions .f. (z),f, ... convergesregularly regularlytotoafunctionf(z) afunctionf(z) which is not not a constant, then any any con·veryes wkick is constant, then of R B contains points z,, such tkat that neighbourhood neig/WourltOOd N zo of qf aa point point z,, z 0 of z,. suck f,. (z,,) (z,.) = f (zo), f,, if n isis sufficiently sufficiently large. If If this this were were not so, so, the given given sequence sequence would would yield an an infinite infinite subsubthat, in in the the neighbourhood N,,,ofof;, sequence, sequence,/,.1 (z),f"' (z), .~., such such that, neighbourhood N,.. z0 , (z), (z),..., (z) .f(z,)) zero. Since the the functions functions (f.~ (z)f (z.)) all differ differ from from zero. the boundary boundary
66
REPRESENTATION THEOREMS OF CONFORMAL CONFORMAL REPRESENTATION
[CHAP. v V
function (1(z) (f(z)f(z function —1(z0)) ofthis thislast last sequence sequencevanishes vanishesatat the the point zz,, 0 )) of 0, Theorem 33 shows showsthat that (f (f(z) —1(z0)) Theorem (z)f (z0)) vanishes identically, i.e. the the funcfunction 1(z) is tionf(z) is aa constant, constant, and and aa hypothesis hypothesis is is contradicted. contradicted.
104. Application to conformal representation. 104. followingtheorem theoremisisofoffundamental fundamentalimportance importanceininthe thetheory theory of of The following conformal representation representation:: conformal Iff,(z), functions which THEOREM. If / 1 (z), /f2(z), sequence of offunctions whick converges converges 2 (z), ... is a sequence regularly domain B, R,and andjfifthe thefu11ftions giveconformal conformal transfortram(orregularly in a domain functions give which are are uni... respectively, whick mations qf of B matiws R into into simple simple domains domains SlJ 8S2, 21 ••• formly bounded, then, either either the the boundary boundaryfunction functionf 1(z) bounded, then, (z) is aa constant, constant, or it gives gives a conformal cMiformal tran:iformatioo simple domain domain S. transformation of RB into a simple S. By hypothesis, f,. (z) f,, (z0) z0. hypothesis,/,. (z) *=Ff,. (z0) when z and andzz,0 are are points points of of B, R, zz =Fz •. The functions 4>,. (z) ==f,. (z0) donot notvanish vanishininthe the pricked pricked (punlctiert) (punktiert) (z) — f,. (z 0 ) do f,, (z)domain R—.z0. Thesefunctions functionsconverge convergecontinuously continuouslyinin this this domain, R z 0 • These domain, By Theorem Theorem 33 of § 103 103 either either towards the the function function41(z) 4> (z) =1(z) =f(z)—f(z0). f(z0 ). By 4> (z) isisidentically then aaconstant, constant,oror414> (z) (z) isis different different 41(z) identicallyzero, zero,andf andf (z) is then from zero zeroand andsof(z)=Ff(z.). sof(z) from 105. The main maintheorem theoremofofconformal conformalrepresentatIon representation (21). 105. The (21). Let B R be be an an arbitrary arbitrarybounded boundeddomain domain in in the the zplane, zplane, containing containing the point z = and therefore therefore also a circular area K defined by =00 and (105i) Iz I < f' ...... (105'1) interior. No No assumption assumption is made made as connectivity of in its interior. as to the connectivity of R. B. Consideraafamily family{f {f(z)} offunctions functions which which are are regular regular in in the circle Consider (z)} of circle (105'1). The The family familyisis assumed assumedtotobe bemade madeup upofofthe thefunctionf(z) functionf(z) 0 1(z) which satisfy the following conditions: and also all functions functionsf(z) following conditions:
=
(a) .f(0)=0, f(O) =0, analytic continuation continuation of possible along off(z) 1(z) is is possible along every every path path y'Y (b) analytic B and I (z) within R and the thefunction function/ (z) isis always always regular, regular, two paths paths joining joiningzz==0 (c) if y' 0 to the points z' and z" y' andy" and )/' are two (z") are are the the values values obtained obtainedat.at z'I respectively, and (z') and and '('F (z") respectively, and if if .,.F .,F (z') z" along these these paths, paths, then then ifif z' z' =Fz" and z" z" by by continuingf(z) continuingf(z) along .,.F(z')=F.,.. F(z") ..... (105"2) provided z#0), (in particular, .,F (z) =F 0 provided z =F 0}, (d) with with the the above above notation notation I.,F(z)l< 1. 1. ...... (1053) (105'3)
106. ItItwill willfirst first be be proved proved that that the thefamily family{f(z)} {f(z)}isiscompact compact(§ 100). 100). Condition (d) shows showsthat that any any sequence sequence of offunctions functions of ofthe thefamily family conta.ins contains
§§ 104107] 104—1071
67
THE MAIN MAIN THEOREM THEOREM
a subsequence subsequence 11 .it (z), (z), /122 (z), (z), ... which satisfies the relation lim ...... (1061) (106"1) urn /,, (z) =fo (z)
,._..,
K. We have to show show that that 10 fo (z) belongs to the the family family {f(.:)}. in x. so that either It is obvious obvious that/ (0) = 0, so either condition condition (a) (a) isis fulfilled fulfilled or It is that!00 (0) f(z)n f(z) 0. 0. must show that ifif y'y' isis aa path path within B To verify condition (b) (b) we must show that within R point z', z', and and ifif for for every every point point { of y', y', other z', other than z', joining z = =00 to aa point (z) gives a function .,.F0 which exists, the analytic continuation of fo (z) yFo which is regular and can be be obtained as the limit of the analytic continuations .,.F,. ('> of the (z), then all these conditions conditions are satisfied satisfied at the functions f,. (z), the point point z' z' itself itself and and in in aa certain certain neighbourhood neighbourhood of that point. point. It isis easy (z) are are regular regular in in a It easy to to show show this, this, for for the the functions functions 1F,. yFn (z) certain neighbourhood neighbourhood of of z' and bycondition by condition (d) (d) they theyform a normal family certain which converges ,,F0 (z) (z) at at all all points of a certain portion converges to iFo portion of of yy' (§ 102). To prove condition (c), (c), consider considertwo twopaths pathsy'y' andy" and y" with with distinct distinct endz", and and let Nz' and Nr· be nonoverlapping neighbourhoods points z' and z", neighbourhoods respectively. By Theorem Theorem 4, 103, there points z0' z,.' in of z' z' and and z" respectively. 4, §§103, there are points Nz' and z,." in Nz'' such such that the equations equations . (1), 1F,, 1F,. yFn (zn') (z,.') = = 1F0 yFo (z'), y"F,.(zn") (z,.") == 1F0 yFo (z") ...... (106"2) hold simultaneously, being suitably chosen. chosen. By simultaneously, nn being By hypothesis hypothesis the terms these equations are unequaL unequal. The 'fhe required required result, result, on the left in these yFo (z') * y"F', (z"), follows. Finally, obvious that/ (z) satisfies satisfies condition condition (d). (d). Finally, it is obvious that!00 (:)
=
m
Consider now now aa particular functionf(z) 1(z) of of the the family and its 107. Conf?ider particular function family and analytic continuations yF(z} in B. R. Suppose Suppose that thatthere thereisisaanumber numberw0, w0 , where 8 w0=he'°, w (OO. f,. ( I ) = 1, I,.' (I ) > 0.
. ..... (124•2) (1242)
The general theory shows (.z) converges converges continuously shows thatf,. that!,, (z) continuously to to zz in in the halfplane. This can that halfplane. can be be verified verified at once; once; for for calculation shows shows that f,. (z) =
1) + ..jf+n2( 1). 21(z + z ~ z:z
(1243) . ..... {124·3)
(c) Let (c) Let the the wplane wplane be be cut cutalong along an an arc arc of of the the unitcircle uni~circle joining the points e"•tn and ei"1", and of — 1/n). of length length 2n271" (1 (1A simplyconnected simplyconnected domain is thus formed, domain is formed, with the point point w w = a:; as as an an interior interior point. point. Let the interior interior of of the the unitcircle, unitcircle, Iz I is transformed transformed conforinally conformally into r~"> in the the same same plane, plane, by by means means of a function function if! (z,.) with ifr if! ((0) 0) ==00 and 1/r' (0) (O) >>0, 0, and that that the thesame samefunction function transforms transforms 0~" 1> into the the domain domain 1 r~ >. Then Then the thetwo twooriginal originaldomains domains may maybe be replaced replaced by by their their images images ri:—'). affectingthe the problem problemjust juststated. stated. From without materially affecting From this itit follows follows that there in the assumption that the there is is no no loss loss of generality involved involved in assumption that domains o are the the circles circles n
lz,.l>0. 1 (0) = and can be determined so as t.o to then /'~"! 1 (0) is proportional to (0} tor,.+., andr.. +, determined so as ensure ensure that
f("l (0) = 0 n+l '
f,(n) (0) = n+l
1.
...... (12.5'3)
We take and require (1253) shall shall hold for all take r1 r 1 = 1, 1, and require that conditions ~onditions (125'3) hold for values of of n.n. Then values '11hen the the numbers numbers r,. are all uniquely uniquely determined. determined.
§§ 126128]
79
SIMULTANEOUS TRANSFORMATIONS SIMULTANEOUS TRANSFORMATIONS
126. The system of equations (n) (z) =f(n+l) (f(n) (z )) f n+2 n n+2 n+l " ' (n) (z,.) J¥,.(z,.)=l/!(¢1"l(z,.)) = (Zn)) (n=1,2, ...)) ...... (130"3) give conformal j Zn conformal transformations transformationsofofthe thecircles circles Iz,.II < r,. into domains which lie within iIt I n
at the the centre centre z0 z0 and domain R. Here nn isis aa positive positive at and lying lying outside outside the domain R. Here By means means of of rotations rotations about about z0 z0 integer. By integer. through angles angles 27r 4ir 2(n—1)ir
n'_tz_' ... •••''
the domain domain R R gives gives rise to to new new domains domains
R1, 14, ...,, R,.1respectively. The R2, ... Thecommon common ... R,._ 1 of all part RR1R2 RR1R2 ... all the thedomains domains the point is an an open open set set which which contains contains the point Pig. zo but which which has no point point of of the the circle circle Fig. 27 27 z0 has no z—z0I=r =r as asan an interior interior point point or or as as aa frontierpoint~ frontierpoint Among Izzo Among the the domains whose sum makes makes up this common whose sum common part there there is is one, one, 14, R 0 , which which contains the point z0. — zz00 I
136. Lemma ~. 2. 136. Considertwo twocuts cutsrw' y,,,'and and',,,," intothe the interior interior of of the domain Consider rw" into domain B,,,. Rw. We suppose that that they We suppose they join join the point point Ow to the the points points w' w' and w" of the
§§ 136, 136, 137]
85
JORDAN DOMAINS DOMAINS
frontier and have have only onlythe thepoint point 0., 0,,,in in common. common. The The result result just just proved shows that thatVw' y.,' and ye" y.," are images of are the images of two twocuts cutsinto into the the interior interior of B2. R •. These These cuts cuts y,,' y,' and y." join the point point o. to the points points C rand(;' and C" of the Jordan Jordan curve curve c. c. We Wenow nowinvestigate investigatenecessary necessary and sufficient sufficU!nt conconditions ,, and r' should slwuld coincide. ditio,zs that C' curve (ye' (yz' + +yz") 14, into into two domains R 1 and R'!l. If(! The curve ye") divides R. If C' and C' r' coincide, then, since ccis curve,one oneof of the the domains, domains, say say R, and is a Jordan Jordan curve, is such that all points of "tz' or of y." or of both. all its its frontierpoints frontierpoints are points both. This domain corresponds to a domain, domain, B R':] say, say, which which may have have points corresponds to the frontier of R., B,,, as as frontierpoints, frontierpoints, but but such a frontierpoint o(j) of B R~> of the cannot be at aa positive distance from the curve (y,,,' + positive distance from the curve (y.,' y.,''). For suppose that such a point. that ww is such point. There 'rhere isisaaneighbourhood neighbourhood N., of ww such that if w,, w,,W2, w2 , ... ••• is is a sequence sequence of of points points converging converging to to aa point point of of the the frontier frontier 1(z), say of R"j R::!within within14,. N.,,, then the inverse of the function functionf(z), say 4, q, (w), takes 4, (w) By § 133, values converging tor. By§ (w) is constant; butthis thisisisimpossible. impossible. to constant; but 133, cp has On the other other hand, hand, if B R':] property in in question, question, so so that, in On the property w", then rC' == '"· particular, w' This is is readily w' == w", readily proved proved if the the above above C". This argument is applied to the transformation w==1(z) and the domain transformation w f (z) and domain argument is applied to
!,
R'~,. z
137. Transformation of (23). 137. Transformation of one one Jordan Jordandomain domaininto intoanother another (23). assumethat that the frontiers of both the We now now assume the domains domains 14,, R., and 14, Rz are Jordan curves, denoted by by c,,, c., and and c,.. C2 • Let w w be an .Jordan curves, which which may may be denoted to,,, ... denote an arbitrary arbitrary point arbitrary pointofofc,,, c., and and let letto1, w1 , w arbitrary sequence sequence of of 2 , ••• points of R., tending tendingto tow. w. The corresponding pointsofofB,. R.are arez1, z 1 , z,, z;., .... corresponding points Since c., c,,.isisaa Jordan Jordan curve the points points w1, w,, w,,, w2 , ... ••• may be joined, joined, each to its itssuccessor, successor, by byaasequence sequenceof ofarcs arcswhose whoseaggregate aggregateforms forms aacut cuty,,, y 111 from ww into the the interior interiorofof11w. R.,. C of Cz into the By § 135, 135, yy,,, theimage image of of aa cut "t• from some point By§ point' 111 isisthe ...,,so so that interior of B,,. R •. Also, Also, y. contains the the sequence sequence of of points points z1, z1 , z,,, z2o ... of the sequence this sequence sequence must must converge converge to to C' '· The convergence convergence of sequence is simply simply aa consequence of the convergence {z,.} to C 'is consequence of convergence of {w,.} to w. w. It It follows that the C to converges depends only follows that the point point' to which which {C,.} converges only on on w, w, not choice of the thesequence sequence{w,.}. {w,.}. A A point point CCcorresponds uniquely to to on the choice each point since R. I?, and pointo,, w, and, since and B,,, R., may be interchanged without without modimodification corresponds uniquely uniquely to to each the reasoning, reasoning, aapoint pointww of c,,, c., corresponds fication of the point ofc,,. c•. pointCCof w22 of c.., c,,, have have two It is is seen seen at at once once that thattwo twodistinct distinctpoints pointsw1 w1 and w distinct corresponding andC2 C2 of ofc,.. c.. (This also follows follows from from distinct correspondingpoints pointsC1C, and
86
TRANSFORMATIONOF OF THE THE FRONTIER TRANSFORMATION
[cwtP. VI [cHAP. VI
§§ 136.) Further, Further, the transformation transformation of other is is of one one frontier frontier into into the other continuous. continuous. For Forlet letw1, ~, ~ •... to w0, w 0 , and let w2, ... be points of Cw converging to pointsofof;. ~~> '2> ... be be the corresponding corresponding points c•. that the If klc is aa positive If positive integer, integer, aa point pointwk wk can be be found in in Rw such that relations 1
1
are satisfied satisfied simultaneously simultaneously by by wk and and its its image image Zk. z,.. But the the points points w to w0. w 0 • It also follows that that Z1o ~ •... ...,, and therefore also w1, w2, ... converge to It follows 1, w 2 , ••• ...,, tend to a point the points ~I>~ •••• point~w2and whichare arepassed passedthrough through w1, w Finally, let let~. w 3 be three points points of ofc,,, cw which 2 and w3 in this this order orderwhen when the thecurve curveisisdescribed described in in the the positive positive sense. Suppose Suppose that cuts into the interior of Rw join Ow to the thatthree threecutintotheinteriorof the respective respectivepoints pointsw1, w~> "'s and w3, w 3 , and that these these cuts do do not not intersect intersect one oneanother anotherexcept exceptatatOs,,. Ow· We consider that the consider the the corresponding corresponding figure in the zplane and observe that the is conformal conformal at Ow· Then 'fhen it is clear dear that the the points points~~. transformation is occur on on c. c2inin this this order order when when the the curve curve isis described described in in the ~2 and ' 3 occur positive sense. These results results may may be be summarized summarized as as follows: follows: into another, another, THEOREM. coriftYrmally into THEOREM.IfIfone oneJtYrdan Jordan domain domain is traniftYrmed ccenformally the transformation isis oneone oneone and continuous continuo1ts in the the closed closed domain, domain, then the and and the the two tu'o frontiers frontiersare aredescribed describedininthe thesame samesense sense by by aa moving moving point point one and and the corresponding on one ctYrresponding point on on the the other. other.
of this this theorem theorem isis easily easily made. made. Let Rw 138. A slight generalization generalization of domain, and c,, Cw a a Jordan Jordan curve curve(with (with ororwithout withoutits itsend endpoints), points), be aa domain, conditionsare aresatisfied: satisfied: (a) every point and suppose suppose that the the following following conditions c,,,can canbe be joined joined w of c,, c,., is frontierpoint of Rw, (b) (b) every point point ww of Cw is a frontierpoint (c) every to any by aa cut into any interior interior point point 0,,, Ow by into the the interior interior of of Rw, (c) Jordan w1w2 c,, and and two .Jordan domain domain whose whose frontier frontier consists consistsofofa aportion portion w1 w2 of Cw two cuts into into the theinterior, interior,O,,w, O,w1and andO,,w2, O'"w2 , lies entirely within within Rw. Then Rw JtYrdan curve. curt•e. 14k,isissaid saidtotocontain containa/ree aires Jordan Suppose that w, § 136 showsthat that the cuts Suppose w1 =!= w Then § 136 shows cuts Ow~ and • (02. 2 two cuts cuts 0.'1 and O.Co, where ' 1=1= * ( 2 • The transO,w 2 are the images of two formation the interior of the formation ww==f(z) f (z) then transforms transforms the the Jordan Jordan domain domain ofof thethe Jordan domain OWWIW2OW. § 137, 0.,1 { 2 0. into into the theinterior interior Jordan domain Oww1w2 0 10 • By By§ 137, any arc any arc of of the the free free Jordan Jordan curve curve c.,., and hence hence the the whole whole curve, curve, is is continuous image image of of an an arc arc of of the frontier c. of B,,. a oneone oneone continuous R •. Just as not the image as in in §136 § 136 ititmay may be be shown shown that that c" .is not image of the whole frontier of R. except when c,,, the whole whole frontier frontier of B,,,, c, isisthe Rw, i.e. when when Ru is a Jordan Jordan domain. domain.
§§ 138, 139]
87
INVERSION
139. Inversion with respect to an analytic 139. anal:vtfc curve. A curve. in the xyplane :cyplane is given either by by an an equation equation A real real analytic curva F (x, y) = 0 ...... (139"1) (1391) or in parametric form form by two equations y=ifr(t). . ..... (139"2) (1392) X=0. > 0. The Thefunction function if! (t) i'l uniquely determined Further, if the these conditions conditions (§ 112). Further, the two two figures figures are are inverted inverted by these with respect to A1B1 A 1 B 1 and A,B2 A 2 B 2 respectively, respectively; they are transformed into into themselves. From this itit follows that themselves. follows that ;jr (l) =.; (t), ...... (1411) (141"1) where ~ and Itare are the the numbers numbers conjugate conjugate to if! and t.t.
§§ 140143] fi 140—143]
INVERSION PRINCIPLE INVl!;.RBION
89
The relation (14r1) (141"1)shows shows that thatip.p (t) maybe may be written in the the form form of a powerseries coefficients, so thesegments segmentsA1 A 1B1 B 1 and and A2 A 1 B2 B2 powerseries with with real coefficients, so that that the correspond to one correspond to one another. another. Hence R~' and Rc' are corresponding corresponding domains.
142. To Toobtain obtainaaconformal conformal transformation transformation of of R~' into the interior interior of z z,, z~o z2, ... ...,, lying within the
I
f(z) such that j 1 ~~ tending to z = 1, such ~z) and, by (1(151"2), 1lf(z,.)l 1 — 1lz,.l are bounded; the the theorem theorem of of §§93 93 can can then then be be used. used. let zz1, ;, ••• In the latter case, case, let arbitrary sequence sequence of of points points ... be an arbitrary 1 , z2o within ABC = ':1.. Let two ABC tending tending to to zz= two sequences sequences of of numbers, numbers,rr,, r,, ... 1 , r~o defined by by the relations and Pi, Pt, P2, ..., ... , be defined and cLr,, 1~z.. <Xr,. ( ) . r,.=k', Pn=1 p,.=1,·,.(ltX)' ······ (1513) 151"3 where tX has the same same meaning meaning as in (93"10). Further, let
1z=r,.(1t), ...... (1514) (151"4) 1 —1(z) = p,. (11/(z) ,. (t)} (t)} is a normal family, so also (1515) with (t)}, and (t,.) = 1. Now Now differentiate differentiate (151"5) with is {4>,.' (t)}, and that lim tj>,.' (ta) for t. This respect to t,t, and, and, taking taking account account of of (151 (151"4), substitute t,. for respect 4), substitute givesf'(Zn) (z,.) r,. = p,.tj>,.' (t,.). Thus (ta). Thus gives!' limf' (z,.) =ex. . ..... (15r9) (151"9) 0
0
0
•••

,._..
theorem:: We have proved the the following following theorem thefunetionf (z)is regular and 11(Z) THEOREM. 1, tkefunctionf(z)isregularand lf(z) II (u) which transforms u 1 I Assume that that rr,._ 1 has already been represented and that that the the figure figure ( T,._ 1 + T,.) has been been represented represented on on aa domain domain of of a appears in both representations, and if if the vplane. triangle T,._1 appears representations, and vplane. The triangle image of a point of T,._ 1 in one is made to correspond to the im&ke of the same point The same point in in the other, a new conformal conformalrepresentation representationisisset set up. up. The is thus reduced representation representation of rr,. on on Izz I