An Expansion of Meromorphic Functions

VOL. 18, 1932

MATHEMATICS: J. L. WALSH

165

Summary.-Crossing-over data from a homozygous III-IV translocation in Drosphila melanogaster indicate that the spindle fibre interferes with crossing-over in its immediate neighborhood. Acknowledgments.-The writer is grateful to Professor T. Dobzhansky who suggested the study here reported and furnished the original stock carrying the translocation studied. Professor A. H. Sturtevant has made many helpful suggestions for which the writer is thankful. 1 NATIONAL RESEARCH COUNCIL FELLOW in the Biological Sciences.

Dobzhansky, T., Genetics, 16, 629-658 (1931). Dobzhansky, T., Ibid., 15, 347-399 (1930). Sturtevant, A. H., Carnegie Inst. Wash. Pub., 421, 1-27 (1931). 6 Muller, H. J., and W. S. Stone, Anat. Rec., 47, 393-394 (1930). 6 Dobzhansky, T., Proc. Nat. Acad. Sci., 15, 633-638 (1929). 7 Morgan, T. H., C. B. Bridges, and A. H. Sturtevant, Bib. Genet., 2, 1-262 (1925). 8 Dobzhansky, T., Amer. Nat., 65, 214-232 (1931). 9 Sax, Karl, Journ. Arnold Arboretum, 11, 193-220 (1930). 10 Janssens, F. A., La Cellule, 25, 387-411 (1909). 1 Darlington, C. D., Proc. Roy. Soc., B107, 50-59 (1930). 12 Belling, J., Univ. Calif. Pub. Bot., 16, 311-338 (1931). 2

3 4

AN EXPANSION OF MEROMORPHIC FUNCTIONS By J. L. WALSH DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY

Commmunicated January 9, 1932

It is the primary object of this note to exhibit a development in series of an arbitrary meromorphic function, a development which represents the function throughout its entire domain of definition, and which can be chosen in an infinite variety of ways. THEORE;M I. If f(z) is a meromorphic function of z (that is, analytic except possibly for poles, at every finite point of the plane), if all the poles of f(z) belong to the sequence a,, a2, a3, . . > 0, where every pole occurs a number of times at least corresponding to its multiplicity, and if the numbers 13, 32, 13, *** are distinct from the a, and uniformly limited, then there exists a unique expansion f(z) = ao + a, X - °f1 + a2 (Z2- ai)(z- a2) +

valid for all finite values of z other than the points ai. The series (1) converges absolutely for all values of z other than the ai and uniformly in any

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PROC. N. A. S.

finite closed region containing no point aj. In such a region the convergence of (1) is more rapid than that of any geometric series. In particukar we may choose #I = 0, (3m+I = 1/a-. Then if f(z) is analytic for l zl _ 1 and if I am I > 1, the sum of the first n + 1 terms of (1) is the rational function of degree n whose poles lie in the points ai1, a2, . . ., a,,, of best approximation to f(Z) on C: z = 1 in the sense of least squares. If f(z) is analytic at z = 0 and we choose 3,i = 0, ai $ 0, then the sum of the first n + 1 terms of (1) is the rational function of degree n whose poles a which has contact of highest order with f(z) at lie in the points a,, a2, ..n. the point z = 0. = 0, 1m+i = r2/am am | > r, and if f(z) is Similarly, if we choose O3' analytic for z < r, then the sum of the first n + 1 terms of (1) is the rational function of degree n whose poles lie in the points ao, a2, ..., a,,, of best approximation to f(z) on z = r in the sense of least squares. If the function f(z) is an entire function, the development (1) is valid for all values of z or for all values of z except the an, according as we have an = 00 or a. - aco. In the former case, (n = 0 gives both best approximation in the sense of least squares on an arbitrary circle z = r, and contact of highest order at the origin If we have an = co (3,, = 0, the expansion (1) is the Taylor expansion of f(z) about the origin. Even though the values a, = co and (, = co cannot be used in (1), according to a literal interpretation of the formtula, we interpret that formula as admitting the use of those values-although in Theorem I itself the value ,,n = co cannot occur. Moreover, we do not suppose the an or the On all distinct, but suppose merely that no a,n shall equal a (Bi. If the (,, are all distinct, an expansion (1) of a given function valid in the points (n is unique. For if we set z = ,I we have the condition

I I

I

I J

aO =

f(&l).

If we now set z = (32, we have f((2)

=

ao + al

(2

-

so that a, is determined. In a similar manner we see that all the coefficients an are successively uniquely determined. This same fact (uniqueness of the determination of the coefficients an as a result of the validity of (1) in the points (n) also holds even if the 3,, are not all distinct, provided we make the convention that the validity of (1) at a point ( at which k of the points (3, coincide shall be taken as equivalent to the representation of f((3) f,((), . ., f(k-1) (() by (1) and the respective derived series of (1). Under this convention, the coefficient a, is determined as follows. Let ao, al, ..., a,,1 be known and let precisely k of the numbers (1., 83*-*., (3,, coincide with O3n+1. If we differentiate (1) formally k times

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and set z = + we see that a. is uniquely determined by the resulting equation. Another way of expressing the formal relationship between f(z) and the a., by assuming (1) valid in the points 3,, is to determine a. or an+k after ao, a,, ..., a,, -1 have been determined, by the relation

(z-a,)(z-a2)...(z-a.)) F (zX )(Z - 2)- (z -.) Zf(z) (X

2)..* . (Z - a2)*** (2 -

____ ao-a

-

-

- 1(Z - a)(z

+ afl+2( +2(Z

-

i

+

a-1)j

cn + l ) (Z 3,Dz

a,,

z-D1Z a,2+ Z -

. an+ 2) (3,)+*

(2 (2)

and this is an expansion of precisely the form (1). If the left-hand member of (2) is not defined for a particular value z = k it is to be defined as the limit of that left-hand member as z approaches pk. With this definition and under the hypothesis of Theorem I, the left-hand member of (2) is analytic even in the points (3k. It is a matter of indifference, as the reader will verify, whether a.+k is determined from (1) by the method previously described, or from (2) by that same method. Theorem I is a special case of the following more general theorem, which we shall also prove. THEOREM II. If f(z) is meromorphic for z < T, if the poles of f(z) of modulus less than T belonig to the sequence cat, C2, . . . , which has no limit point of modulus less than A, and where each pole occurs in the sequence at least a number of times corresponding to its multiplicity, and if finally the points 3I, 12, . . . are distinct from the a,, and have no limit point of modulus greater than B < A, T. then there exists an expansion found by interpolation in the points 1,

I

f(z)

x-1

+ aa

-2) + (Z1-)(Z (z a)

(3) a,) (z valid for z < (AT - BT - 2AB)/(A - B + 2T), z distinct from the points a.. The series (3) converges absolutely for all values of z (other than the a,,) in modulus less than (AT - BT - 2AB)/(A - B + 2T) and converges uniformly in any closed region interior to the circle z = (AT BT - 2AB)/(A - B + 2T) and containing no point a,,. The special cases A = co and T = o are not excluded; Theorem I is the special case A = co, T = . Theorem II is to be proved by means of the following theorem.' THEOREM III. Let the function f(z) be analytic for z < T and let the numbers a,,(i = 1, 2, ..., n; n = 1, 2, . . .) have no limtit point whose modulus = ao

+

a,

z

-

al

-

I

I

I I

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MA THEMA TICS: J. L. WALSH

PROC. N. A. S.

is less than A, and the numbers fi3'(i 1, 2, ..., n + 1; n = 1, 2, ...) no limit point whose modulus is greater than B < A, T. Then the sequence of rational- functions (necessarily unique)

+ .,. a 2l...( ,)(4) (Z °aln) (Z °a2n) (Z ann) whtich coincide with the function f(z) in the points f3in approaches the limit f(z) for Iz < (AT - BT - 2AB)/(A -B + 2T), uniformly for I z < R < (AT-BT-2AB)/(A -B + 2T). If f(z) is not analytic for z < T but is meromorphic, with a finite number of poles of modulus less than T distinct from the Oi3n, Theorem III may still yield a result on the expansion of f(z), as we proceed to indicate. Let the first m of the numbers a,,, a2nY . . . aa,n be ca,, a?, ..., am, independent of n (n > m), and let the poles of f(z) of modulus less than T be included in this set, each pole enumerated according to its multiplicity. Let the first m' of the numbers A,ln. (2n, ... (3n+l,n be i31, (2, * (m' also independent of n(n m'- 1), f3, $ aj. We assume the points am+k,n(k > 0) to have no limit point of modulus less than A and the points 3m'+k,n (k > 0) to have no limit point of modulus greater than B, and f(z) is conveniently assumed analytic in all the points of interpolation Oi3,. Let R(z) be a rational function of z of the form

f(z)

=

+ anlz aonz (z-a~(

+

...

Ml

aoz

1

+

aiz

m'-2

+ ..+

am'

(z - a,)(z -a2) (z. am), which takes on the same value as f(z) in the m' points f i, .2 this function R(z) is uniquely determined. The function

F(z)

=

(Z-C31)(Z .32)...(Za3m )

(3m';

z f(z) -RR(z)]

is then analytic for Izi < T. We approximate to F(s) according to Theorem III by rational functions of the form bozn-m' + b1nz -m'-1 + . + bnm"n F.(z) -(Z am+l,n)(Z am+2,n) ... (z ann) In the case m > m', this function F&(z) is precisely of form (4), where m - m' of the (formal) poles are restricted to lie at infinity. In the case m' > m, this function Fn(z) is of form (4) with the additional requirement F.(oc) = 0 tbis last condition given m' -m times, equivalent to requiring a zero of F(z) at infinity formally of order m' - m. The function Fn(z) is required -

-

-

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MATHEMATICS: J. L. WALSH

to coincide with F(z) in the points f3m'+l,n i3m'+2,n) ... (3,+1,,, and is thereby uniquely determined. In the case m > m' it follows directly from Theorem III and in the case m' > m can be proved in a manner similar to the proof of Theorem III (loc. cit.) that we have lim F.(z) = F(z)

uniformly for z I . R < (AT - BT -2AB)/(A - B + 2T). Then if we set

fn(z) = R(z) + (Z - oi)(z

-

Om') (Z- 31)(Z - 32) ... (Zail)(Z a2) *.. (z am) F.(z) aonz + aj,z`' + *- + a. (z

a2)

...

(z

-

(5)

-

-

-

am)(z - am+,n)

...

(z -a.)

we have

(6)

f(z)= n+lim fn(z) w

uniformly in any closed region interior to the circle z = (AT - BT 2AB)/(A - B + 2T) and containing no point a,, a2, ..., am. The function fn(z) defined by (5) is of degree n, coincides with the function f(z) in the points (31, (32, ... 3m'" Om'+l,nP O3m'+2,nt ... #3n+,,.n and has its poles in the points cal, a2, .P. am, atm+1,nxam+2,n ... a..n. We have thus proved a theorem on the expansion of meromorphic functions, which the reader will readily state explicitly, and which includes Theorem III as a special case, dependent on the hypothesis that the first m of the numbers acil, a2n, .... a., are independent of n (n > m) and that the first m' of the numbers , (32k, ..., + are independent of n(n >- m' - 1); on the other hand, we do not here require, as in Theorem III, that the points ciin have no limit point of modulus less than A and that the points pin have no limit point of modulus greater than B so far as the an and (3n themselves are concerned. In our discussion of development (6) we have for convenience supposed that thefirst m of the numbers aln, a2np ..,. ann ate independent of n(n > m), but it is naturally sufficient if any m of the numbers aln a2n, . . . X a... are independent of n, for n greater than some M. A corresponding remark holds with regard to the numbers (win. In either Theorem III or development (6), if the points a,, (i < n) and 3in (i . n + 1) do not depend on n, the formal development of f(z) is precisely of type (1); for the function fj,(z) - fjn(z) is a rational function of z of degree n which vanishes in the points z = (31, (32, ... (3,,and which has the (formal) poles z = al, a2, * a*:n 9

I

fn (Z)

f-1 (Z)

a,,

(Z -31) (Z

-

(z- al) (z

-

32)

...

(Z

-

3,)

a2)

...

(Z

-

a,n)

170

MA THEMA TICS: J. L. WALSH

PROC. N. A. S.

Reciprocally, a formal development of type (1), where the coefficients are determined as we have described, is a formal development of type (4) where the equations i< n+ 1,

a] = a,ns,

i < n,

are valid; the function fn(z) [sum of the first n + 1 terms of (1)1 equals f(z) in the points (3ix. The same convention that we have made for Theorem I is naturally assumed to hold for Theorems II and III and development (6) relative to interpolation in multiple points (i3,,. Theorem II can now be proved either (i) by identifying (3) with (6) or (ii) by applying Theorem III directly as was done in the proof of (6). Let us indicate the details according to method (ii). Choose arbitrarily A' < A, T' < T, B' > B, and choose N so that for n > N we have an > A', T'; | An| < B'. All the coefficients an in (3) are uniquely determined, and for n > N the left-hand member of (2) is analytic for z < T'. Moreover, the points ain (which may here be written a,,, n > N) have no limit point of modulus less than A', and the points (d,, (which may here be written (3,, n > N) have no limit point of modulus greater than B'. Then for a fixed n greater than N the equation (2) is valid, by Theorem III, uniformly for z < R < (A'T' - B'T' 2A'B')/(A'-B' + 2T'). It is now allowable to let A', T', B' approach A, T, B, respectively; the validity of (2) and therefore of (3) follows, R < (AT - BT - 2AB)/(A - B + 2T). For uniformly for |Iz the absolute convergence of the series (2) and hence of (3) we refer to our detailed proof of Theorem III (loc. cit.). In Theorem III, and hence in Theorem II and expansion (6), if we have #B,, = O, #in = 1/a-j1,., withA, T> 1,thelimit(AT-BT- 2AB)/ (A - B + 2T) can everywhere be replaced by (A2T + T + 2A)/(2AT + z A2 + 1); if the given function f(z) is analytic 1, the approximating functions are the respective functions of form (4) of best approximation to f(z) on Z = 1 in the sense of least squares, provided I a,4 >|. It may be noticed that in Theorem II and development (6) it is not necessary to assume f(z) analytic in all the points of interpolation 3,,. Indeed, so far as concerns the points 3,n of modulus not less than T, the condition fj((3,) = f(3,,) has no necessary relation to f(z) as a monogenic analytic function, and those conditions are essentially arbitrary auxiliary conditions. For instance, we may choose k points 3,n at infinity and write fj( oD) = 0 as a k-fold condition, so that expansion (6) includes the extension of Theorem III used in studying the convergence of F,(z) for the case m' > m in the proof of (6).

I I

I

I I

forI I

VOL. 18, 1932

MATHEMATICS: J. L. WALSH1

171

Series of form (1) have been previously studied by Angelescu,2 but apparently only for analytic (not meromorphic) functions, and only under the hypothesis of the existence of lim a,, and lim- P.. The connection of such series with approximation in the sense of least squares, as used in Theorem I, seems first to have been pointed out by the present writer.3 An expansion of form (1) can readily represent the function zero in a region (not containing all of the points (3,) without having all of the coefficients a. to vanish, as Angelescu2 has indicated. This follows easily likewise from our present discussion. For instance, let f(z) be analytic z < T, and let Pi, 2,2 . . ., P. be of modulus not less than T, while Pn+lp tOx+2, ... are of modulus not greater than B. If the numbers al, a2, ..., are all of modulus greater than A, the coefficients ao, a,, a,,-, in (2) may be chosen arbitrarily; the expansion (2) is valid, as follows from Theorem III, and leads to an expansion (1) for f(z) also valid uniformly for I z < R < (AT - BT- 2AB)/(A - B + 2T). In particular, the expansion is valid if f(z) vanishes identically; the coefficients ao, al, . . ., ax-, are arbitrary. By an arbitrary linear substitution

forl I

I

w

=

as+

5

,y '

cz a

-Ae 0 O,

the expansion (3) is transformed into an expansion of precisely the same form. In general, the- convergence to f(z) at z of an expansion (3) as used in Theorem III depends on the analytic or meromorphic character of f(z) in all points t such that we have I (t, (tya)zg0) _~° t, x, A3)I = { (t-_ a)((-

(Z-a) (t-j3

1,

where a and ,B represent respectively arbitrary limit points of the sets a, an (,d,. Corresponding to this fact, a more general theorem than Theorem III can be expressed, which, moreover, is in a form invariant under linear transformation; compare the reference already given. The restriction a, $ (3k, which we have constantly made, is a matter of convenience rather than necessity. Many of our results are still valid if that restriction is removed. 1 Walsh, Trans. Amer. Math. Soc., 34 (1925). 2 Angelescu, BuU. Acad. Roumaine, 9, 164-168 (1925). 3Compare, however, Malmquist, Compi. rend. du sixime congras (1925) des mathAmaticiens scandinaves, Copenhagen, 1926, pp. 253-259.