Theory of Functions: Part 2 Applications and Continuation of the General Theory [Transl. from the 4th German Ed., Reprint 2021] 9783112399361, 9783112399354

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THEORY OF FUNCTIONS PART TWO

THEORY OF F U N C T I O N S By

DR. K O N R A D K N O P P Professor

of Mathematics

at the University

of

Tubingen

Translated b y FREDERICK BAGEMIHL, M. A. Instructor m Mathematics at the University of Rochester

PART T W O APPLICATIONS AND CONTINUATION OF THE GENERAL THEORY

NEW YORK DOVER

PUBLICATIONS

1947

First American Edition Translated From the Fourth German Edition COPYRIGHT

1947

By DOVER BOOK PUBLISHERS, INC.

PRINTED BY

IN

THE

UNITED

THE W I L L I A M RICHMOND,

STATES

OF

BYRD PRESS, VIRGINIA

AMERICA INC.

CONTENTS PAGE

Introduction

vn SECTION' I

Single-valued Functions C H A P T E R 1 . E N T I R E FUNCTIONS

§1. Weierstrass's Factor-theorem

1

§2. Proof of Weierstrass's Factor-theorem

7

§3. Examples of Weierstrass's Factor-theorem. .

22

C H A P T E R 2 . M E R O M O R P I I I C FUNCTIONS

§4. Mittag-Lefller's Partial-fractions-thcorem. .

34

§5. Proof of Mittag-Leffler's Theorem

39

§6. Examples of Mittag-Leffler's Theorem

42

CHAPTER 3.

P E R I O D I C FUNCTIONS

§7. T h e Periods of Analytic Functions

58

§8. Bimplv Periodic Functions

64

§9. Doubly Periodic Functions; in Particular, Elliptic Functions

73

V

vi

CONTENTS

SECTION

II

Multiple-valued Functions CHAPTER 4.

R O O T AND L O G A R I T H M

PAGE

§10. Prefatory Remarks Concerning Multiplevalued Functions and Riemann Surfaces.. §11. The Riemann Surfaces for v z and log z... §12. The Riemann Surfaces for the Functions w = V(z — a^){z — a 2 ) • • • (z — ak) CHAPTER 5.

ALGEBRAIC

93 100 112

FUNCTIONS

§13. Statement of the Problem

119

§14. The Analytic Character of the Roots in the Small 121 §15. The Algebraic Function CHAPTER 6.

T H E ANALYTIC

126

CONFIGURATION

§16. The Monogenic Analytic Function.

135

§17. The Riemann Surface

139

§18. The Analytic Configuration

142

BIBLIOGRAPHY

147

INDEX

148

INTRODUCTION The foundations of the general theory of analytic functions were laid in Part I of this Theory of Functions.1 Special functions (such as e', sin z, log z, Vz, and others) or classes of functions (such as the rational or the entire functions) were dealt with there only occasionally. Now such more detailed investigations will come in greater measure to the foreground. Only ohce again, later on, will more general considerations be carried out, in order to clarify the situation left undiscussed in I, §24, pp. 103-104. In doing so, it will become apparent that the distinction between single-valued ^nd multiple-valued functions which was indicated there is quite fundamental. This distinction will therefore serve from the outset as a standard for all of the following presentation. From these two main classes we shall select several especially characteristic and important types of functions. A certain arbitrariness is unavoidable in this connection, since completeness within the close compass of this little book is naturally denied us. We shall get away from this danger most easily if we start with the elementary functions (the entire and fractional rational functions, e', sin z, log z, Vz, • • •) as the most theory of Functions, Part I: Elements of the General Theory of Analytic Functions, translated from the 5th German edition, New York, 1945,—referred to in the following, briefly, as "I", together with paragraph or page number. vii

viii

INTRODUCTION

important ones, and try to understand that which is essential and of a universal character in their principal properties.1 The entire rational functions (polynomials)—evidently the simplest and most transparent functions—are characterized (cf. I, p. 137) "purely function-theoretically" by the fact that they are regular in the entire plane and have a pole at the point If one ignores the last property, one arrives at the more general class of entire functions, which are characterized solely by the property of being regular in the entire plane (excluding oo), and to which the entire rational and the entire transcendental functions belong as special cases. They also appeared to us in I, §27 to be the simplest, because their power-series expansion for an arbitrary center converges, and therefore represents the function, in the entire plane. Since analytic continuation, then, is out of the question, the entire functions are naturally singlevalued. In their totality they are identical with the totality of everywhere-convergent power series of the form 0»

g(z) =

£

n-0

and, as such, appear to be an immediate generalization of the entire rational functions. l Thus, we can be concerned in the following with a selection only—with samples, so to speak. The theory of functions is a large realm, which cannot be explored on a journey of one or even several days. If, in spite of this, we undertake on the following pages to sketch briefly a few of the principal places in this realm, we must emphatically caution the reader not to identify the extent of this little volume with that of the theory of functions.

INTRODUCTION

ix

In the first chapter we shall approach these functions with the question: Which of the fundamental properties of the entire rational functions does the class of entire functions still possess, and which not?—and shall give several answers to this question. According to I, §35, Theorems 1 and 2, the fractional rational functions are completely characterized from the purely function-theoretical point of view by the fact that they have no singularities other than poles in the entire plane and at the point m would be convergent, and always with t h e same value zero. T o exclude these cases we employ t h e more useful definition above, and, if necessary, draw attention t o t h e restriction it contains by adding: "in t h e stricter sense".

PRQOF o r W E I E R S T R A S S ' 8 FACTOR-THEOREM

9

The following theorems are easily proved for such convergent infinite products: Theorem 1. A convergent product has the value zero i f , and only i f , one of its factors vanishes. Theorem 2. The infinite product (1) is convergent i f , and only i f , having chosen an arbitrary a > 0, an index n0 can be determined such that I u„+1-un+2 • • • un+T - 1 | < £ for all n > n0 and aU r > 1 (cf. I, §3, Theorem 4). Since on the basis of this theorem (let r = 1 and n + 1 = v) it is necessary that lim u, = 1, one usually p—»00

sets the factors of the product equal to 1 + c„ so that instead of dealing with (1) one is concerned with products of the form (2)

n

(i+c,).

For these, then, c, —* 0 is a necessary (but by no means sufficient) condition for convergence. We make use of the following Definition. The product (2) is said to be absolutely convergent if ft (1 + I c. |) converges.1 'The definition which first suggests itself: "IT«, shall be called absolutely convergent if I I | u, | converges," is not to the purpose, since then every convergent product would at the same time converge absolutely.

10

E N T I R E FUNCTIONS

We then have Theorem 3. Absolute convergence is a sufficient condition for ordinary convergence; in other words, the convergence o / n ( l + | c, I) implies that o / n ( l + c,). On the basis of this theorem it will be sufficient for our purposes to have convergence criteria for absolutely convergent products. The following two theorems settle completely the question of convergence for these products: Theorem 4. The product 11(1 + y,), with y, > 0, is convergent if, and ofily if, the series 2-y, converges. Theorem 5. For n(l + c,) to converge absolutely, it is necessary and sufficient that 2c„ converge absolutely} The following theorem is analogous to one on absolutely convergent series: Theorem 6. If the order in which the factors of an absolutely convergent product occur is changed in a completely arbitrary manner, the product remains convergent and has the same value.2 CO

'According to this, J ~ [ ( l — (z2/j m, all r > 1, and all z in (this is possible by I, §18). Then Fm(z) is actually regular and distinct from zero in ©'. Indeed, if, for n > m, we set fl

(1 + /,(*)) = Pn and Pm = 0

f — m +1

for abbreviation, we have Fm(z) = lim P„ t

n—»oo

= lim [(P m + 1 -P m ) + ( P m + 2 - P m + 1 ) + • • • + ( P „ - P „ - 0 ] , n—»00

•The proof will show that there axe only a finite number of factors in question.

PROOF OF W E I E R S T R A S S ' S F A C T O R - T H E O R E M

13

or (6)

Fm{z) =

E

(P, -

v—m+ 1

P.-0,

and Fm(z) is thus represented by an infinite series. Now the theorems of I, §19 bring us rapidly to our goal. Since, for n > m, \'Pn | < (1 + | /.•,(*) |) " " * (1 + | /„(*) I) < gl/m +•

00

, | 2, I

+00

Consequently, it is possible (indeed, in many ways) to assign a sequence of positive integers kuk2, • • • , k„ • • • such t h a t (3)

is absolutely convergent for every z. In fact, it suffices, e.g.1, to take k, = v + a,- For, no matter what fixed value 2 may have, since zv —• 0 arbitrary, but fixed) is established as follows : Since the series (3) also converges for z = R, and since 2, —» co j m can be chosen so large that 0

0

ir

(5)

< |

and

1 < |

for all v > m. Let us for the moment replace z/z, by u, k, by k, and a, by a. Then, for v > m, the vth term of the series (4) has the form

with {' « J < j W

|