Elliptic Functions: A Constructive Approach 0471965316, 9780471965312

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Elliptic Functions

Elliptic Functions A Constructive Approach

Peter L. Walker

Sultan Qaboos Universi(y, Sultanate of Oman

JOHN WILEY & SONS

Chichester · New York · Brisbane · Toronto · Singapore

Copyright © 1996 by John Wiley & Sons Ltd, Batlins Lane, Chichester, West Sussex P019 IUD, England National 01243 779777 hltemational (+44) 1243 779777 e-mail (for orders and customer service enquiries): [email protected]. Visit our Home Page on http://www.wiley.co.uk or http://www.wiley.com All Rights Reserved.No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any fonn or by any means, electronic, mechanical, photocopying, recording or otherwise, except under the tenns of the Copyright Designs and Patents Act 1988 or under the tenns of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London, UK WIP 9HE, without the pennission in writing of the publisher.

Other Wiley Editorial qfjices John Wiley & Sons, Inc., 605111ird Avenue, New York, NY 10158-0012, USA Jacaranda Wiley Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1LI, Canada Jolm Wiley & Sons (Asia) Ptc Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809

Rritisl, Library Catafogui11g i11 Publicatio11 Data A catalogue record for this book is available from the British Library ISBN

O 471 96531 6

Produced from camera-ready copy supplied by the author using TeX Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn This book is printed on acid-free paper responsibly manufactured from sustainable forestation, for which at least two trees are planted for each one used for paper production.

... they are always interested in technical details when the main question is whether the stuff is literature or not (don marquis)

Contents Preface . . . . . . 0 Preliminaries . 0 .1 Differentiation 0.2 Connected Sets 0.3 Sequences and Series 0 .4 Sequences of Functions . 0.5 Infinite Products .. . 0.6 Integration . . . . . . 0.7 Exercises for Chapter 0 1 Circular Functions 1.1 Eisenstein Series 1.2 Addition Formulae 1.3 Infinite Products 1.4 An Alternating Series 1.5 The Exponential Function 1.6 Power Series . . . . . . . . 1.7 Logarithms . . . . . . . . 1.8 Inverse Circular Functions 1. 9 Geometry . . . . . . 1.10 Asymptotic Estimates 1.11 Some Alternatives . . 1.12 Exercises for Chapter 1 2 Gamma and Related Functions 2 .1 Euler Series . . . . . . . . . . 2.2 The Gamma Function as an Infinite Product 2.3 The Gamma Function as an Integral 2.4 Beta Integrals . . . . . 2.5 Asymptotic Estimates . . . . . . .

Xlll

1 1 2 3 5 7 8 10 11

13 15 18 19 20 22 23 27

31 35 36 39 43 43

46 49 52 54

viii

CONTENTS

2.6 2.7 3

Fourier Transforms . . . Exercises for Chapter 2

62 66

Basic Elliptic Functions 3.1 Lattices ........ 3.2 Basic Elliptic Functions 3.3 Addition Formulae ... 3.4 Weierstrass Functions 3.5 Exercises for Chapter 3

69 70 71 77 80 84

4 Theta Functions . . . . .

Phi and its Translates Theta Series . . . . . . Sums of Squares ... Addition and Rotation Formulae Fourier Series . . . . . . Exercises for Chapter 4

87 87 91 93 95 98 102

5

Jacobian Functions . 5.1 Alternating Series 5.2 Addition Formulae 5.3 Derivatives and Sums of Squares 5.4 Fourier Series . . . . . . 5.5 Exercises for Chapter 5

105 106 112 114 115 118

6

Elliptic Integrals .... 6.1 The Simple Pendulum 6.2 Glaisher's Notation .. 6.3 Elliptic Integrals ... 6.4 The Length of an Ellipse . 6.5 The Amplitude and the Modular Angle 6.6 The Arithmetic-geometric Mean 6.7 Exercises for Chapter 6

121 122 126 131 136 138 141 146

4.1 4.2 4.3 4.4 4.5 4.6

7 Modular Functions ...

7.1 7.2 7.3 7.4 7.5 7.6 8

Modular Groups ... Fundamental Regions The Elliptic Modular Function Other Modular Functions Summary and Special Cases Exercises for Chapter 7

Applications 8.1 Waves ..

151 151 153 156 161 164 168 171 171

CONTENTS 8.2 8.3 8.4

8.5

Entire Functions Some Number Theory Elliptic Curves . . . . 8.4.1 Elliptic Curves over C 8.4.2 Elliptic Curves over R 8.4.3 Elliptic Curves over Q 8.4.4 Finite fields . . . Exercises for Chapter 8

ix

177 180 188 188 191 195 203 207

References .

211

Index . . . .

213

List of Figures 1 2 3 4

The function e and its inverse The function s and its inverse The function t and its inverse Sine and cosine . . . . . . . .

5 6

Zeros and poles off, g and h. Zeros and poles of /14, /24 and

7 8 9

12 13 14 15

The simple pendulum . . . . . . . sn, en and dn on the real axis . . . Relief of the doubly-periodic function i0.877... . . . . . . . . . . . . . . . Relief of the doubly-periodic function i0.877... . . . . . . . . . . . . . . . Relief of the doubly-periodic function i0.877 ... . . . . . . sn and I when 3t, = 0 . . sn and I, general case . . sn and I, modified version Convergence of (dn)

16 17 18

Mapping by S. . . . D 1 and D2 partitioned The mapping by ,\ of D2 onto H.

154 156 158

19 20 21 22 23 24

Interaction of two solitons. . Values of a. . . . . . . . . . Rectangular elliptic curves. Singular elliptic curves. Subsets of 3t, = 0, 3t, = 1/2. Rhombic elliptic curves. . . .

176 187 193 194 196 200

10 11

25 28 30 33

/34.

108 116 122 128 snu for k = 0.8, , = . . . . . . . . . . . . . 129 cnu for k = 0.8, , = . . . . . . . . . . . . . 129 dnu for k = 0.8, , = 130 133 135 136 143

Preface The theory of elliptic functions is one of the high points of classical analysis whose results, in addition to their intrinsic elegance, continue to find uses in fields as diverse as number theory and fluid mechanics. The earliest contributions from Euler, Gauss, Eisenstein and Jacobi were derived to a large extent by direct manipulation of series and integrals; later writers from Weierstrass onwards made their results part of a deeper and more general theory of meromorphic functions in the complex plane. We adopt the former method, strongly influenced by Eisenstein and the monograph of Weil. We shall show how the principal results about elliptic functions can be derived using relatively modest analytical machinery. We begin with circular or trigonometric functions since, as he himself demonstrated, Eisenstein's point of view 1s best appreciated when we follow the analogy between the simple series 00 1 and

~(x+n)k

which define the circular and gamma functions, and the double series

which define the elliptic functions. We shall consider elliptic functions of Jacobi type (usually denoted sn, en, dn, etc.) as well as those of Weierstrass type ( denoted p, (), and elliptic modular functions. In addition to the monograph of Weil already mentioned, we have consulted the books of Chandrasekharan, Remmert, Apostol and Whittaker & Watson among many others which are listed in the bibliography. Most of our results appear as formulae and this has allowed us to replace the 'theorem-proof' style of exposition with a freer form in which each result grows naturally out of the ones before. All readers (including those only casually browsing) should at least read section 6.1 which shows how the Jacobian functions arise from a simple

XIV

PREFACE

mechanical problem and at the same time shows how our approach is related to the classical one. The final chapter which shows how elliptic functions relate to other parts of mathematics can also be sampled at an early stage. The exposition of the main ideas, in which the circular functions are first defined by infinite sums of rational terms, then developed into power series, shown to satisfy addition and duplication formulae, and finally related to infinite products and to simple geometrical constructions, all takes place in chapter 1. These topics are recapitulated in chapter 2 with the difference that the series are summed over N instead of Z. In addition chapter 2 contains sections on asymptotics and Fourier transforms which will be needed later. The development of these ideas as they apply to double series and products occupies chapters 3 to 6; without understanding this structural relationship between chapter 1 and the later chapters there is a real danger of being submerged by the flood of formulae. Chapter 3 develops the ideas of sections 1.1 and 1.2 on Eisenstein series, deriving nonlinear relationships between them and finding addition and duplication formulae for elliptic functions which are analogous to the corresponding results for circular functions in chapter 1. In addition, these double series satify rotation formulae which have no analogue in the onedimensional case. Chapter 4 on theta functions extends similarly the infinite products of section 1.3, and chapter 5 on Jacobian functions grows out of the short section 1.4 in which alternating signs are introduced. In chapter 6 we study elliptic integrals, which allow the inversion of elliptic functions in the same way that the sine function is inverted by an integral in section 1.8. Chapter 7 is devoted to functions which depend only on the lattice constant r and studies their mapping and transformation properties. The final chapter shows how elliptic functions relate to other parts of mathematics. This chapter considers (i) the solution of the KdV equation by solitons and cnoidal waves, (ii) the role of modular functions in proving results about entire functions, (iii) some applications in number theory to the theta-multiplier system, and (iv) an introductory treatment of elliptic curves, currently one of the most intensively studied areas of mathematics. The reader should of course be prepared from time to time to check the details of a calculation even when given in full in the text, since it is only in this way that one gets any confidence with the material. Indeed mathematics has in common with poetry that it is difficult simply to read from the printed page; a higher degree of involvement is required. (It has to be said however that the conventional advice to those who find poetry difficult, to 'read it aloud' has very little effect on mathematics.) On the other hand, once one has written out a number of the derivations in detail (from section 3.3 for instance) the methods become familiar and the purpose clear. Anyone wishing to find further material need only look at the classical literature-for instance Whittaker & Watson-to find many more results on which to practice.

PREFACE

xv

To avoid overburdening the text, some quite substantial material has been put into the exercises; usually these are 'headed,' as for instance exercises 4.12 on zeta functions, 6.11 on the Landen transform, and 7 .10 on division points. Comments in parentheses () are intended to be simple amplifications of the text. Comments in brackets [] are intended rather as statements of how the material might be viewed from a more advanced viewpoint. We use the useful convention f := g to signify that the expression g is to be taken as the definition of f. It is a pleasure to acknowledge valuable contributions from colleagues: Dr. E. V. Krishnan on waves, Dr. Y. Abu-Muhanna on entire functions, and Dr. M. D. Ha for help with the exercises. Dr. Bachir Yallaoui gave expert advice on the preparation of the typescript. Several anonymous referees drew my attention to important references and suggested improvements in organisation, and David Ireland of J. Wiley's editorial staff was unfailingly helpful. The book is dedicated to a class of Omani students, with whom I rediscovered (not for the first time) the pleasures of establishing attractive and non-trivial mathematics with a minimum of prerequisites.

0

Preliminaries This preliminary chapter sets out the rather limited background in analysis which we shall need. We assume that the reader is familiar with the basic facts about open, closed and compact sets in R (the real line) and C (the complex plane). We also assume the notion of continuity and its consequences for the mapping of compact sets. We begin with the definition of differentiation, not so much because it is likely to be unfamiliar, but to emphasise what is and is not being assumed. Later sections consider convergence of infinite series and products of complex functions, and the integration of a continuous real or complex valued function on an interval. The Cauchy integral theorem and its consequences are not needed except for applications in the final chapters.

0.1

Differentiation

Suppose that f is a real or complex valued function defined on an open subset D of R or C. For a point c E D we say that f is differentiable at c if the quotient (f(c + h) - f(c))/h tends to a limit as h - 0 in D. The limit, if it exists, is denoted by f' ( c) and is called the derivative of f at c. When the limit exists at all points of D we have a function f' on D (also called the derivative off) and we say that f is differentiable on D. As is well known we can linearise the definition by saying that f is differentiable at c if there is a complex number a = f' (c) for which we can write f(c+ h) = f(c) +(a+ c)h where c - 0 ash - 0 in D. We say f is holomorphic on D when it has a continuous derivative on the open set D C C .. [The requirement that the derivative is continuous is not usually imposed but is automatically satisfied in the examples we shall consider; in any case the general Cauchy-Goursat theory shows that it is a consequence of differentiability on an open set in C .] We say that f is analytic

2

ELLIPTIC FUNCTIONS

at c if there is a power series with positive radius of convergence whose sum is f in some neighbourhood of c. Power series are considered in more detail in section 0.4 where we show that an analytic function must be holomorphic; the converse implication will not be needed. A function which is holomorphic on C is said to be entire. From elementary analysis we assume the mean value theorem: if f is real valued and differentiable on an interval ( a, b) C R and is continuous on [a, b] then for some point c E ( a, b) we have f (b) - f (a) f'( c)(b- a). In particular if f' = 0 on (a, b) then f is constant there. The mean value theorem is false for complex functions as is shown by the example a= O,b = 21r, f(x) = eix_ Consequences of f' being zero on an open set in C are considered in the next section. We shall assume the elementary rules for differentiating products, quotients and composite functions, which remain valid for both real and complex functions.

=

0.2

Connected Sets

An open set G in C is connected if it satisfies one of the following equivalent statements: (i) if G = G 1 U G2 where G 1 n G2 = 0 and both G1 and G2 are open, then either G 1 or G 2 must be empty, (ii) every continuous function from G to Z is constant, (iii) every pair of points a, b E G can be joined by a segmental path in G, i.e. for some finite set of points {a = zo, z1, ... , Zn = b} C Geach [zj-1, Zj] C G. (Here and throughout, if a, b E C then the line segment from a to bis defined as the set [a,b] := {z = (l-t)a +tb: 0::; t::; 1}.) To show the equivalence we begin with (i)--+(iii). Let a E G and let G 1 be the set of points z of G for which a segmental path from a to z exists and G 2 be the rest of G (if any). Since G is open it is easy to show that both G1 and G2 are open and they are obviously disjoint. Hence by (i), since G 1 is not empty, G2 must be. To show (iii)--+(ii), suppose that f is a continuous function from G to Z which is not constant so that for some a, b E G, f(a) -:f. f(b). By (iii) there is a segmental path from a to b and the restriction Ji off to this path gives a continuous function from [O, 1] to Z with fi(O) -:f. fi(l) which contradicts the intermediate value property. Finally to show (ii)--+(i) we see that if (i) fails then G G 1 U G 2 where G 1, G 2 are disjoint nonempty open subsets of G. But then the function f (z) = j for z E Gj, j = 1, 2 is continuous and (ii) fails also. For connected sets in R the definition and proofs are the same, and the third statement shows that the only connected open subsets in Rare the open

=

PRELIMINARIES

3

intervals. The situation for non-open sets is more complicated; fortunately we shall not need to consider it. A nonempty connected open set will be called a region in C. As consequences of connectedness, we note the following results which will be useful later. (A) If f, g are functions which are continuous and non-zero on a region G and f 2 = g2 on G then either f = g at all points of G or f = -g at all points of G. (B) If f is a holomorphic function on a region G and f 1 = 0 on G then f is constant on G. The proof of (A) is immediate from (ii) of the definition since the ratio f / g is continuous and equal to ±1 at all points of G. For the proof ~f (B), it is enough by (iii) to prove that f is constant when f' = 0 on a segment 1 [a, b] C G. But f(b) - f(a) = f'(z)dz = 0 J'(a + t(b - a))(b - a)dt = 0, where we have used the definition of the integral of a function along a segment in C from section 0.6.

J:

0.3

J

Sequences and Series

A sequence is a functions : N-+ C, where we write Sn for the value of sat n, and call it the n th term of the sequence. The sequence is convergent with limit s if for all E > 0, there exists no such that lsn -sl < E when n ~no.We assume the elementary properties of convergent sequences such as linearity and the uniqueness of the limit. We write (sn) or (sn)o to specify the sequence s. A series is determined by taking a sequence ( an)o and defining a new sequence (sn)o by Sn = L~ am, We call (an) the terms of the series and ( sn) the partial sums. We say that the series is convergent with sum s if the sequence Sn has limit s, and in this case we write I:~ an = s. Again we shall assume the convergence of examples such as the geometric series I:~ zn for lzl < 1 and the sum of powers I:~ n-k for k > l, and the simplest forms of the ratio and comparison tests for convergence of more general series of positive terms. There are obvious modifications of these definitions when the sequence or series has terms which are defined only for sufficiently large values of n, n ~ n 0 say. We shall often come across series of the form L~oo an. They are normally regarded as the sum of two series, namely I:~ an and I:::::~ an = L~ a_n, and are said to be convergent when both converge. However we shall see that it is often useful to consider grouping terms to obtain a0 + I:~(an + a_n), the so-called Eisenstein convention. This is not equivalent to the customary definition of convergence since it does not even require that the terms should tend to zero; for instance the series L~oo n has sum zero by the Eisenstein

ELLIPTIC FUNCTIONS

4

convention. Other more typical examples are found throughout chapter 1. If the series is convergent in the normal sense, then use of the Eisenstein convention will result in a convergent series with the same sum. We say that L~ an is absolutely convergent if L~ Ian I is convergent. An absolutely convergent series must be convergent but the converse need not be true as the example :Z::::~(-lt /n shows. We assume the comparison and integral tests for absolute convergence, namely that we may conclude that an is absolutely convergent if either (i) L~ bn is a convergent series of positive terms, and for some no,A we have Ian I ::; Abn for n 2 no, or f (x) dx < oo and (ii) f is a positive decreasing function on (X, oo) with Ian I ::; f (n) for n 2 X. We say that an is unordered convergent with sum s if for each c > 0 there is a finite set S 0 C N such that for all finite sets S :::) So, I LnES an - sl < c. It is a pleasant but not entirely trivial exercise to show that a series of complex terms is absolutely convergent if and only if it is unordered convergent and that the sums are the same; the reader who does not wish to prove this can find it in [Wa2]. Having established the equivalence we may deduce in particular that the terms of an absolutely convergent series may be rearranged in any way without affecting either the convergence or the value of the sum. A valuable feature of unordered convergence is that it applies equally to sums over any index set, and remains equivalent to absolute convergence. In particular it applies to double series of the form am,n and shows that if the double series is absolutely convergent then the sum is independent of any rearrangement of terms, and in particular is independent of the order in which the summations are taken. We illustrate this with the following example taken from section 1.2. When the series

L~

J;

L~

L~ L~

00

m~oo """'

1

00

"""' - - - - - - n~oo

(x

+ n)2(y + m -

n)2

x, y (/:.

z

first occurs it is summed as indicated first over n then m. If we consider the series of absolute values

:z=:=-oo 1/IY + m - nl

summed first over m, then the sum n and so the whole summation reduces to

2

is independent of

PRELIMINARIES

5

where both series are clearly convergent. Hence the original series is absolutely convergent and may be summed as in section 1.2 by interchanging the order of the summations. To end this section we investigate circumstances in which we may take limits inside a summation. To be more explicit we consider convergent series of the form Sn = I::=l am,n where for each m, am,n -+ tm as n -+ oo and we ask when the limit Sn exists and is given by L~ tm if this is convergent. That this is not generally true is shown by the example of am,n = 1/n for 1 :S m :S n, am,n = 0 otherwise. Then Sn = 1 for all n while tm = 0 for all m, and it is equally easy to devise other ways in which the result may fail. A simple condition ensuring the required result is that there exists Am > O such that I:~ Am is convergent and lam,n I :S Am for all m, n. The seque~ce Am is said to dominate am,n and the result to be proved is one of the many versions of the dominated convergence theorem which we shall see again in section 0.6. To prove the result, notice that since am,n -+ tm as n-+ oo we have ltm I :S Am and so I:~ tm is absolutely convergent with sum t say. Given f. > 0, choose M such that I::+i Am < c/4, whence I::+i ltml < c/4 also. Then choose N such that lam,n - tml < c/(2M) for n 2 N and m = 1, 2, ... , M. Then we can write Sn - t = L~ (am,n - tm) + L:+i(am,n - tm) and so for n 2 N we have lsn - ti :S M c/(2M) + 2c/4 as required.

0.4

Sequences of Functions

It usually happens that the series which we consider have terms which are functions, Sn sn(z) say, where z belongs to some subset S of C. It is natural to say that Sn ( z) -+ s( z) as n -+ oo if for each z E S and f. > 0 there is n 0 EN (depending on z and c) such that lsn(z)-s(z)I < f. when n 2 n 0 ; in this case we say that the sequence is pointwise convergent on S. However this is not always the most useful form of convergence to consider since for instance the sequences given by (i) sn(x) = xn, x E [O, 1], (ii) sn(x) = -./x 2 + n- 2 , 2 X E [-1, 1], (iii) Sn ( X) = n x(l - X X E [O, 1] are all pointwise convergent on the indicated intervals, but in (i) the limit is discontinuous, in (ii) it is continuous but not differentiable and in (iii) the integral Sn does not tend

=

r'

J;

to

J01 s as n -+ oo. This leads to the stronger notion of uniform convergence. We say that

Sn -+ s uniformly on S if for each f. > 0 there is n 0 E N such that lsn(z) - s(z)I < f. for all n 2 no and z E S (now the same choice of no works for all points of S). We assume the following results which show that uniform convergence is sufficient to avoid the problems illustrated above: (i) if each Sn is continuous on S and Sn -+ s uniformly on S then s is continuous on S.

ELLIPTIC FUNCTIONS

6

(ii) if each Sn is differentiable on Sand both sn(zi)-+ l for some z 1 ES and -+ ef> uniformly on S, then Sn is uniformly convergent to a differentiable functions on S with l = s(z1) and¢= s'. (Notice that in example (ii) above, Sn is uniformly convergent but s~ is not.) (iii) if each Sn is continuous on a bounded interval [a, b] and Sn -+ s uniformly on [a, b] then J: sn(t)dt -+ J: s(t)dt as n -+ oo. The same is true more generally for integrals over a path in C; we shall consider such integrals in section 0.6. We shall frequently meet series with rational terms such as L~oo 1/(x + n) which we shall interpret as 1/x + L~ 2x/(x 2 - n 2 ) using the Eisenstein convention. This series is absolutely and uniformly convergent for all x E C \ Z as a consequence of the Weierstrass M-test which is an extension of the comparison test to series of functions: if there exists a convergent series L~ Mn with positive terms such that lan(z)I ::S Mn for all n and z E S, then the series L~ an (z) is absolutely and uniformly convergent on S. When we consider multiple series such as Lm n (x + m + nr)-k (~r > 0) in chapter 3 we shall use the M-test, compari~g the terms with those of Lm n ( m 2 + n 2 )-kf 2 • For this we need the following estimate which is valid for ;ufficiently large m and n.

s~

1/(2 + lrl) Jm2 + n 2

< -


0) and divergent for lz - cl > r (if r < oo). We also assume that if f (z) is defined as the sum of L~ an (z inside its circle of convergence ( assumed to have positive radius) then f has, in the same circle, derivatives of all orders given by j(k)(z) = L~=k n(n - 1) · · · (n - k +

1

Cr

Cr

PRELIMINARIES

7

l)an(z-ct-k, and so in particular that f(k)(c) = k!ak for all k. This is easily proved by expanding f(z + h) in powers of h, using absolute convergence to justify the necessary rearrangements. The series I:~ f(n)(c)(z - c)n /n! is called the Taylor series of f at c. Moreover f is analytic at all points of N(c,r) := {z : lz - cl < r}, that is if d satisfies Id - cl < r then f(z) = L~ bn(z - dt where bn = JCn)(d)/n!, and the series has radius of convergence of at least r - Id - cl; this too is proved by a simple argument using rearrangements. A much deeper fact about analytic functions is that if not constant then they are open mappings, that is the image of any neighbourhood of a point c at which f is analytic is a neighbourhood of f(c). The proof uses some facts about complex logarithms and is given as exercise 1.9. In addition to Taylor series we shall also meet Laurent series in which a finite number of terms with negative indices are allowed, so that we have series of the form I::=-N an ( z = g( z) say. Such a series can evidently be written in the form (z - c)-N f(z), where f(z) = I:~ an-N(Z is a Taylor series as already considered, so that if the Taylor series is convergent for lz - cl < r then the corresponding Laurent series is convergent for O < lz - cl < r. If a_N # 0 we say that g has a pole of order N at c. The coefficient a_ 1 is called the residue of g at c. A function which is analytic on an open set G except for a number of poles at points of G is said to be meromorphic on G. We emphasise that for the functions we shall consider, the Taylor or Laurent series come as direct consequences of the definition of the relevant functions, as can be seen for instance by inspection of sections 1.1 and 3.2. The general theory which ensures the existence of such series for arbitrary holomorphic functions is not required.

Cr

0.5

er

Infinite Products

An infinite product is an expression of the form TI~ Pm where the terms Pm are complex numbers, and the value assigned to the product could reasonably supposed to be the limit as n --+ oo of the partial products Pn := TI~ Pm. It turns out that this is overhasty since the presence of a zero term makes Pn = 0 for all sufficiently large n, which is not always what we want; for instance the product in which po = 0, Pn = n! for n ~ l does not correspond to something we should want to think of as convergent. Thus we proceed more cautiously in the treatment of zero terms. If, in the product TI~ Pm, infinitely many terms are zero then we say the product is divergent, irrespective of the behaviour of the other terms. Thus from now on we can suppose that for some N, Pn # 0 for n ~ N. Then we can say that the product is convergent if the partial product TI~ Pm = QM say

ELLIPTIC FUNCTIONS

8

Tir

tends to a non-zero limit Q as M ---+ ex>. The value of Pm is then defined as PoP1 · · · PN-1Q, which will be zero if and only if one of Po, ... ,PN-1 = 0. If Pm ---+ 0 as M---+ ex> then we again say that the product is divergent. Since convergence implies Pn ---+ l as n ---+ ex> it is convenient to write Pn = l + an where now an --+ 0 is necessary but by no means sufficient for convergence. The general theory of convergence of infinite products is slightly delicate (see [Apl], exercises to chapter 12), so we shall restrict ourselves to the simpler situation of absolute convergence. For am > 0, the product TI~ (1 + am) converges if and only if the series L~ am is convergent; to see this notice that PN := TI~ (1 + am) is increasing and so is convergent if and only if bounded above. Then PN 2:: l+ao+· · ·+aN and so if PN converges then so does I: am. Conversely if I: am converges then there is an integer k for which a 0 + · +aN < k and so by exercise 0.7, PN::; 3k; thus if I: am converges then so does PN. If the terms an are complex numbers with I:~ an absolutely convergent then we say that the product (1 + an) is absolutely convergent and it is easy to show that this implies that the product is convergent. Typical examples of infinite products are given by TI~(l - x2 /n 2 ), which occurs in section 1.3 and converges absolutely and uniformly for all x E C, and the double product TI'm,n (l-x/(m+n,)) whose convergence, subject to the Eisenstein convention, is established in section 4.1. It is important to be able to differentiate an infinite product p( z) = TI~ (1 + an ( z)), assumed uniformly convergent. From the results in section (1 + an(z)) is differentiable and 0.4 it is enough to show that PN(z) := that P'tv is uniformly convergent. But P'tv/PN = a~(z)/(l + an(z)) by Leibniz' formula for the derivative of a product. Hence it is sufficient for us to know that if the series I:~ a~ ( z) / ( 1 + an ( z)) is uniformly convergent, for which the M-test is usually sufficient, then P'tv is uniformly convergent and pis differentiable with p1 /p = L~ a~(z)/(l + an(z)). Since the result is equivalent to the formal process of differentiating the series I:~ log(l+an(z)), it is usually referred to as logarithmic differentiation, even though logarithms are not involved in either the statement or the proof.

TI%

n~

Tir

0.6

Lr

Integration

In elementary analysis we show that a continuous function f : [a, b] ---+ R is integrable, and that if F( x) := fax f (t)dt then F is differentiable and F'(x) = f(x) on [a, b]. From this we deduce that if f' is continuous then

J: f'(x)dx = J(b) - f(a).

We shall want to extend this in several ways. The first is that for integrability of f we can allow a finite number of discontinuities, provided

g

PRELIMINARIES

that f remains bounded in a neighbourhood of each discontinuity. Then F remains continuous, and F' = f except at the points of discontinuity. The second extension is to complex valued functions, when for f = Ji + i/2 we define f = Ji + /2. This gives us a class of functions which is sufficient for the discussion of Fourier series and integrals in sections 2.6 and 4.5. We assume the basic properties of integration such as linearity and the rules for substitution and integration by parts, and the estimation rule IJ f I :=:; J If I which is exercise 0.5. We also need to be able to differentiate under the integral sign: if F(x, y) is continuous on a rectangle R := [a, b] x [c, d] and has a continuous first partial derivative F1(x, y) = 8F/8x(x, y) on R, then g(x) := F(x,y)dy is differentiable on [a,b] and g'(x) = F1(x,y)dy. The third extension of the scope of integration is to integrals over paths in C (we need this only for exercise 1.9 and for the applications in chapters 6,7 and 8). A path in C is a piecewise continuously differentiable function 'Y say from a compact interval [a, b] of R to C. Then the integral of a continuous function f over 'Y is by definition

J

J

J

t

t

r f(z)dz

h

:=

1

f('Y(t))'Y'(t)dt.

[a,~

In particular if 'Y is the line segment from zo to z1 m C, then -y(t) zo + t(z1 - zo) and

1

f(z)dz

= (z1 -

zo)

[z 0 ,zi]

/1 f(zo + t(z1 -

lo

zo))dt.

For the basic properties of path integrals one can consult the introductory section of any textbook of complex analysis, for instance [Wal]. One important application of the notion of a path integral is the fact that holomorphic functions have primitives on certain subsets of C. We say that a set Sis star-shaped if for some zo ES, the interval [zo, z] CS for all z E S. A typical example of a star-shaped open set is the complex plane with a half-line removed. If f has a continuous derivative on a star-shaped open set D ( centre zo) then there exists F with F' = f on D. For let F(z) = hzo,z] f(u) du, where

= f01 f(zo +t(z-zo))(z- zo)dt,

[z 0 , z] CD since Dis star-shaped. Then F(z)

so by differentitaion under the integral sign we have

F'(z)

1'

f(zo

+ t(z -

zo))dt +

[1 i_[tf(zo + t(z -

lo

dt

l'

f'(zo

+ t(z -

zo))]dt = [tf(zo

zo))t(z - zo)dt

+ t(z -

zo))]6 = f(z)

as required. As a consequence, for any two points a, b in a star-shaped region = F'(z)dz = F(b) - F(a). Hence

S for which [a, b] C S, we have ha,b] f

ha,b]

f

= ha,c] f + hc,b] f

J:

when all of [a, b], [b, c] and [a, c] are subsets of S.

ELLIPTIC FUNCTIONS

10

0. 7

Exercises for Chapter 0

0.1 Square roots. Let z = x + iy be a complex number with y f:. 0, and let u = J(lzl + x)/2 (the positive square root of a positive real number) and v = y/2u. Show that ±(u + iv) are the two square roots of z. What modifications are needed if y = O?

I:r

0.2 Let f(z) = an(z - at have positive radius of convergence and a1 f:. 0. Show that lf(z1)-f(z2)l 2:'. ia1ilz1 - z2l/2 for z1,z2 sufficiently near to a, and hence that f is 1 - 1 in some neighbourhood of a. 0.3 Let f be analytic on a region D and let A= {z: f(z) = O}. Show that if A has a limit point in D then f = 0 in D. More generally show that if f, g are analytic on D and f = g on A then f = g on D. (First show that f = g in a neighbourhood of a, then use connectedness.) This is the analytic identity principle; see [Re], page 227 for a discussion of this result. 0.4 Prove the results using rearrangements of power series omitted from section 0.4. 0.5 Show that for an integrable function

f : [a, b]

-+

J: lfl. (Consider a complex number t with !ti= 1 and t J:

J:

I f I :s; f = I fl.)

C we have

J:

0.6 g-summation. Let g be a continuous decreasing function on [O, oo) with g(O) = 1 and g(t)-+ 0 as t-+ oo. Let f be a continuous function on [O, oo). We say that f is g-summable with value A if fa°° f (t)g( ct)dt -+ A as c -+ O+. Show that this is compatible with the usual definition of convergence of integrals, i.e. prove that if f (t)dt -+ A as x -+ oo, then also f is g-summable with value A. 0.7 (i) Use the binomial expansion to show that for integers n 2:: 1,

J;

~ n(n - 1) · · · (n - r L-i r!nr

(1 + 1/nf

+ 1)

r=O

n


0 for x ER while sh(x) has the same sign as x for all x ER - {O}, and both ch, sh are increasing on (0, oo ). We may get alternative expressions for the coefficients 'Y2k of E 1 ( x) from these expansions as follows. We have E1 ( x )s( x) = c( x) and so on replacing each function by its power series we find for O < Ix I < 1 that

X [

and

f .

(~l)j

J=O

1 (px)2j]

(2J + 1).

r

[_.!. X

f

'Y2kX2k-ll

k=l

n-l

=

f n=O

(-1)~ (px)2n (2n).

r

·

1 (- 1 2n """" ( -1 ) 2j _ ( -1 2n (2n + l)!p (2j + l)!p 12 (n-j) - (2n)! p '

for n

f;:o

= 1, 2, 3, ... , or equivalently, n-l

·

""""(-1r-J

'Y2(n-j)

f:o (2j + 1)! p2(n-j)

2n (2n+l)!.

(1.26)

For n = 1 this gives the known relation ,2/p2 = 1/3 which was used in section 1.2 to define p 2 . For n 2: 2, (1.26) is a recurrence relation for 'Y 2 k which shows the unobvious fact that each ,2k is a rational multiple of p 2k. The recurrence may be solved successively to give 14 = p 4 / 45, 16 = 2p 6 /945, etc. For historical reasons we write ,2k = (-l)k- 1 (2p) 2kB2k/(2k)! where B2k, k 2: 1 are the Bernoulli numbers which are rational and alternate in sign. Then B2 1/6, B 4 = -1/30, B6 = 1/42 etc. and the recurrence may be written n-l (2n + 22(n-J) B2(n-j) = 2n for n 2: 1. (1.27) j=O 2J + 1

L .

1)

·

Nonlinear recurrence relations for , 2k and B2k can be obtained by substituting the power series for E 1 and E2 into E2 = Et + p 2 . Bernoulli numbers are the values at x = 0 of the Bernoulli functions which are discussed in section 2.5.

1. 7

Logarithms

This section and the next are concerned with constructing inverses to the exponential and circular functions. None of these functions are one to one on C and so in each case we have to be careful to specify appropriate regions on which local inverses can be defined. In each case the construction is by means

24

ELLIPTIC FUNCTIONS

of an integral, in fact the integral of a complex function along a line segment as outlined in section 0.6. The integral which defines the inverse sine will be generalised in chapter 6 to give elliptic integrals. We begin with a careful investigation of the mapping properties of the exponential function e( x). We start by proving (i) e maps R onto the unit circle U := {z: lzl = 1}, and in particular is a bijection from (-½, ½J to U, (ii) e is a bijection from iR to ( 0, oo), and (iii) e maps e onto ex := e - {O}, and in particular is a bijection from (- ½, ½J x R to ex . (Since e has period 1, we can choose any half-open interval of length 1 in (i) and (iii).) For x = t E R we know from (1.23) that e(t/2) = c(t) + ips(t), where c(t), s(t) are real. It follows from c2 + p2 s 2 = l that e maps R into U. In addition we note that for n E Z, s(t) is positive if t E (2n, 2n + 1) and negative if t E (2n - 1, 2n), while c is positive on (2n - ½, 2n + ½), and negative on (2n+ \, 2n +~).Since s' = c, c' = -p 2 s, it follows thats is strictly increasing on [O, 2J with s(O) = 0, s( ½) = 1/p and that c is strictly decreasing on [O, ½J with c(O) = 1, c( ½) = 0. Hence e is a bijection from [O, ¼J to that part of the unit circle x 2 + y 2 = l with x, y 2:: 0 (the first quadrant of U). Similar arguments show that [¼, ½J, [-½, -¼J and [-¼, OJ map bijectively onto the other quadrants of U, and (i) is established. For x = -iu E iR we have the power series e(-iu) = I:~(2pur /n! which is clearly positive and strictly increasing from [O, oo) onto [1, oo ). Since e(iu) = l/e(-iu), e(-iu) is also positive and strictly increasing from (-oo, OJ to (0, lJ. This establishes (ii). For any z f= 0 we can write z = clzl with !cl = 1, and hence from (i) and (ii) there exist t E (-½, ½L u E R with e(t) = c, e(-iu) = lzl. The addition formula now shows that e(t - iu) = z and so e maps e onto ex. The rest of (iii) now follows from (i) and (ii). The complex number w = t - iu, (t E (-½, ½L u E R) which we have just uniquely defined for z E ex to satisfy e( w) = z gives us a function l: ex -+ (-½, ½J x R which is inverse toe on the appropriate domains; that is l(e(w)) = w for all w E (-½, ½J x R, and e(l(z)) = z for all z E ex. The function l is continuous only on the set e- := e - (-oo, OJ -+ x R since it is easily checked that the limits of l( z) as z approaches points of (-oo, 0) from above and below differ by 1. However we have not developed the machinery (the Cauchy-Riemann equations in polar form) that is needed to show directly that l is differentiable. Instead we shall show this by a different route, first constructing another inverse of e by means of an integral. In the introductory chapter we showed how to define the integral of a complex-valued function along a line segment in e by means of its

(-½, ½)

CIRCULAR FUNCTIONS

25

~

l w-plane

Figure 1

The function e and its inverse

parametrisation,

f f(z) dz:= (b J~,~

a)

[1 f(a + t(b - a)) dt.

Jo

We need the case in which f (z) = 1/ z and we integrate along the segment [1, z], which for z E c- does not contain the point O where f is discontinuous. Thus we define

L(z) :=

1

-du =

[l,z] u

1 1

- d l t, - -z o l+t(z-l)

z E

c-.

(Alternatively the second integral may be used as the definition of L and we can dispense entirely with general considerations.) Then by differentiation under the integral sign (or directly, by considering the limit as h -+ 0 of (L( z+h )-L(z)) /h) we have L'(z) = f 0\ 1+t( z- l) )- 2 dt which is equal to 1 if z = l, otherwise

L'

z -

-

l

1

- _l_

()-(z-l)(l+t(z-1)) 10 ,-z-l

(

1-

!) - ! z

-z

and so L'(z) = l/z for all z E c-. For x E D := x R, we know that e(x) E c- and so we may consider the composite function L(e(x)) on D. Its derivative is L'(e(x))e'(x) = 2ipe(x)/e(x) = 2ip, and hence L(e(x)) = 2ipx + c, where the constant c is

(-½, ½)

26

ELLIPTIC FUNCTIONS

found to be 0, by putting x = 0, i.e. L(e(x)) = 2ipx for x E D. This shows that the functions l and L /2ip are equal on D, and also incidentally that l is differentiable. We next consider the power series expansion of L. Since L'(z) l/z, we have L(n)(z) (-l)n- 1 (n - l)!/zn and hence for any c E c- we have the formal Taylor series

=

=

S(z)

= Sc(z)

·-

L(c) +

oo

L (-lf- 1 1

L(c)

+

f

(-1r-1 n

l

The series has radius of convergence

S'(z)

= ~(-lf-1 (z ~ 1

lei

(n -1)1 . (z - cf nl en .

(z - c)n C

and satisfies

cr-1 =

en

1

c + (z - c)

= L'(z)

=

on B1c1(c). But S(c) = L(c) so S(z) L(z) whenever B1c1(c) Cc-, which is true for c -:p O with ~c 2: 0. (Here Br(a) := {z: Jz-aJ < r} is the open disc, centre a, radius r.) Hence S(z) gives the power series expansion of Lon B1c1(c) if c -:p 0, ~c 2: 0. But if ~c < 0 we have a more complicated situation in which B1c1(c) intersects (-oo,O) wh~re Lis discontinuous. In this case S(z) L(z) on that component of B1c1(c)nc- in which c lies (that is the upper if8'c > 0, or the lower if 8-c < 0), but S(z) L(z) ± 2ip in the other- component. The addition formula for e, e(x + y) = e(x)e(y) implies a corresponding result for its inverse provided we are careful about domains. For instance if ~z > 0, ~w > 0 and we put x = l(z), y l(w), then both x, y have real parts in(-¼,¼) from our construction of l. Then ~(x + y) E (-½,½)and e(x + y) = e(x)e(y) = zw, so that x + y = l(zw) and we conclude that

=

=

=

l(zw)

= l(z) + l(w) for ~z > 0,

=

~w > 0,

and of course the same is true for L 2pi l. The result is also valid if one (but not both) of ~z, ~w is only required to be 2: 0. We can use the function l to construct a holomorphic square root function which will be useful to us in the next section. For suppose that z E c- so that ½l(z) E (-¼,¼) x Rand g(z) := e(½l(z)) has positive real part. In particular, if z E (0, oo) then g(z) E (0, oo) also. The addition formula fore now shows that g2 (z) = z for z E c- and so g is a function on c- which gives a holomorphic square root there. More generally, if f is any holomorphic function on a domain D -+ c- then go f is a holomorphic square root off on D. We shall use the traditional notation v7 for g o f, the so-called principal square root of f. Similarly L is the principal logarithm and we shall write

CIRCULAR FUNCTIONS

27

log(!) for L o f when f is a function whose range is a subset of c-. For z E c- the imaginary part of L(z) is the argument of z denoted by arg(z), so that -p < arg(z) < p for z E c-.

1.8

Inverse Circular Functions

We now construct inverses for the circular functions s(x) and t( x) which is defined by

= x Il~(l-x 2 /n 2),

t(x) := s(x) = _1_ = -E 1 (x + 1/2) c(x) E 1 (x) p2 For s we recall from the discussion in section 1.7 that s is a continuous and strictly increasing function from to [-}, } ], which is entire and

[-½, ½]

2

satisfies s' = 1- p 2s 2 on C. Hence if we want to construct an inverse function for s we should consider the integral

I(y) :=

r jl -

dt

Jo

which elementary calculus tells us defines an inverse for s, at least on [- l, l]. Since we are interested in complex values of the argument we must fina tbe values of y E C for which the integral I(y) (now regarded as being taken over the interval [O, y] C C) is well defined. We know from the preceeding section on logarithms that a function has a holomorphic (principal) square root if its values are in c-, so we want to find those values oft for which 1- p 2 t 2 E c-, or equivalently 1 - p 2t 2 (/:. (-oo, 0), p 2t 2 (/:. [1, oo). But p 2t 2 E [1, oo) if and only if t E (-oo, U oo ). Thus we define

-¼] [¼,

1 1 D1 := C - ((-oo, --] U [-, oo)), p p

and note that [O, y] C D1 when y E D1. Hence

I(y)=

1,

[O,y]

dt

-;::===,

jl -

p2t2

is well-defined and section 0.6 tells us that we may differentiate to obtain I'(y) 1/)1- p 2 y2 for y E D1. We see from the addition formula for s in the form s( u +iv) s( u )ch( v) + ic(u)sh(v) (see (1.25) for ch, sh) that S's(x) and S'x have the same sign if x E S := x R where c(u) > 0. Hence if x E S then s(x) E Di, 1 - p2s 2 E c- and so j l - p2s 2( x) is well-defined and holomorphic on S.

=

=

(-½, ½)

28

ELLIPTIC FUNCTIONS

But s' (x) = 1 - p2 s 2 (x) on C, so s'(x)/jl - p2 s 2 (x) is well-defined and equal to ±1 on S. Since S is connected the ratio must be a constant which equals 1 since this is its value at 0. Thus we have s'(x) = Jl - p 2 s 2 (x) for 2

XE

S.

......---... s

~

I

Figure 2

The function s and its inverse

Hence I( s( x)) is well-defined for x E S and satisfies

I(s(x))'

= I'(s(x))s'(x) = Jl -

2 2

p s (x)

Jl - p2 s2 (x)

= 1,

so that I(s(x)) = x for x E S and in particular s is one-to-one on S. To complete the proof that I is inverse to s, we need to show that s maps S onto D1. This is already known for real values so by symmetry we need consider only the subregions of S and D1 with positive imaginary part. Recall that if x = u + iv, then s(x) = q + ir = s(u)ch(v) + ic(u)sh(v). Hence for fixed v = v 0 > 0 the image of the interval {u +iv:-½< u < ½,v = vo} is given by that part of the ellipse

for which r > 0. Conversely any point q0 + i r 0 E D 1 with ro a unique v > 0 with

ro qo ( sh(v) ) 2+p ( ch(v) )2 = l, 2

> 0 determines

CIRCULAR FUNCTIONS

29

since the left-hand side is a strictly decreasing and continuous function of v > 0 which tends to +oo as v-+ O+ (since ro > 0) and tends to O as v-+ oo. Then for this v, the point P := ro/sh(v) + ipq 0 /ch(v) is on the unit circle so there is a unique u E (-½,½)with P = c(u)+ips(u) and so qo+iro = s(u+iv) as required. To obtain the power series for I we use the binomial expansion as follows. For jxj < 1 we have (l+x)

_.!. 2

= Loo n=O

(-1/2) xn n

(-1/2)(-3/2) · · · (-n + 1/2) = Loo ----------xn; n! n=O

this may either be taken on trust, or more in the spirit of this work the reader can verify directly that f(x) := I::=o (-~l 2)xn satisfies 2(1+x)f'(x)+ f(x) = 0, and hence that (1 + x)f 2(x) = 1. We derive a general form of the binomial theorem in exercise 1.6. Hence

I(y)

=

[ J[o,y]

dt

Jl - p2t 2

=[ J[o,y]

! ~ (1/2)(3/2) · · · (n p

~

(2n + l)n!

t

(1/2)(3/2) · · · (n - 1/2) (pt)2n dt n!

0

1/2) ( )2n+l PY '

where the series has radius of convergence l/p. For t we follow a similar procedure except that no difficulties arise with square roots. We have t(x) = l/E1 (x) = s(x)/c(x) and it follows from the discussion of s and c in section 1. 7 that t is a strictly increasing function from (-½, ½) onto (-oo, oo ), which satisfies t' = (s' c - cs')/ c2 = (c2 - p 2s 2 )/ c2 = 1 + p 2 t 2 . Again elementary calculus tells us to consider

J(y) := [

l +du2

J[o,y]

P u

2'

which is inverse to t, at least for y E R. This leads us to define the set D2 := C \ i((-oo, -1/p] U [1/p, oo)); 2 2 notice that 1 + p u f:. 0 if u E D 2 , and that y E D 2 => [O, y] C D2, so J is well-defined and J'(y) 1/(1 + p 2 y2 ) on D 2 • A simple calculation shows that

=

t( u +iv)

=

s( u )ch( v) + i c( u )sh( v) c(u)ch(v)- ip 2 s(u)sh(v)

= s( u )c( u) + i sh( v )ch( v) jc(u + iv)j 2

-½

and hence that if x = u +iv E S, < u < ½, then ~(t(x))/~x ~(t(x))/~x > 0. Note also that the function th which we define by . . sh(v) th(v) := t(zv)/z = -h() C

V

le(-iv)-1

= -p e (-ZV. ) + 1 ,

(1.

28

)

> 0 and

ELLIPTIC FUNCTIONS

30

~

J

Figure 3

The function t and its inverse

using (1.25), is strictly increasing from (-00,00) onto (-1/p,1/p). Hence t maps S into D2 so J(t(x)) is well-defined on S and J'(t(x))t'(x) = (1 + p2t 2)/(l + p2t 2) = 1, and J(t(x)) = x for x E S. In particular, t is one-to-one on S. Again to complete the proof that J is inverse to t we have to show that t maps S onto D2. From (1.28) we have .

t(u+iv)=

t1

+ it2 .

2 l-ipt1t2

wheret 1 :=t(u),t 2 :=th(v),

(1.29)

and if q0 + i r 0 E D 2 is given ( with say q0 > 0 for definiteness) then we want to find u + iv ES with t(u +iv)= qo + iro. From a geometrical point of view we may argue that (1.29) gives a bilinear function of t1 and t2 which for fixed t1 maps the line u = const. in S onto the circle in D2 which passes through the points ±i/p (when t2 -+ ±oo) and t 1 (when t 2 = 0). All points of D2 lie on one of these circles and so D2 is the image of Sunder t. Alternatively we can give a purely algebraic proof by equating real and imaginary parts in qo + iro = (t1 + i t2)/(l - i p2t 1t2) and eliminating to get the quadratic equations

qop 2ti + (1 - p2(q6 + r6))t1 - qo rop 2t~ - (1 +p2(q6 + r6))t2 + ro

0

0.

The first of these has a unique positive root and this determines a unique value of t1 and thus of u. The second has two positive roots whose product is

CIRCULAR FUNCTIONS

31

1/p2 so there is a unique root in (0, 1/p) which gives v uniquely. The power series for J is obtained as before from the binomial expansion.

with radius of convergence again equal to 1/p.

1.9

Geometry

The reader will have been waiting impatiently for us to identify p, defined in section 1.2 by p 2 = 6 I:~ 1/n2, with 1r defined in elementary geometry as the ratio of the circumference of a circle and its diameter. We do this first, following it with a list of formulae for 7r which we have established on the way. Once this has been done, those readers who wish to define circular functions by power series may refer to section 1.6 to make the expected identifications. We prefer to go a little further in the direction of elementary geometry to define the usual radian measure of an angle by means of arc length ( and so as an integral). This gives the geometrical definition of the circular functions as the ratios of lengths, with the added bonus that we have established the infinite product, and power series representations of these functions. We finish the section with a list for reference of the functions we have defined with their names in customary mathematical notation, to which we shall adhere from now on. To begin with we adopt from elementary geometry the name 21r for the length of a unit circle x 2 + y 2 = 1. It follows that the first quadrant where x 2 0, y 2 0 has length 7r /2. From section 1.7 we recall that this quadrant is the bijective image of the interval (0, 1/2) under the map t ~ (c(t),ps(t)). We also recall that the length of such a curve is given by (or is defined as, if one prefers that point of view) the integral

[

Jx'(t) 2 + y'(t) 2 dt,

for a general curve given by t ~ (x(t), y(t)), a :=; t :=; b. In our case we have x'(t) = c'(t) = -p 2 s(t), y'(t) = ps'(t) = pc(t) and so the required length, which we know to be 7r /2, is given by /1/2

lo and we have p

,------

jp4 s2 (t)

/1/2

+ p2 c2 (t) dt = p lo

= 1r as hoped for.

dt = p/2,

ELLIPTIC FUNCTIONS

32

This gives us the following representations for 1r established so far:

6

1 7r

2 7r

7r

f~

which is our definition of p.

n

1

00

s(l/2)

= 2 IT 1

f

4

n1 2 )

,

or

since c(l/4)

f (-+r 0

(Wallis' product).

1 · 3 .. · (2n - 1) 1 2,4 .. ·(2n) 2n+l·

0

4

1-

~ . 3 · 5 ... (2n - 1)(2n + 1) 2·2 4 ·4 (2n ) 2 1/2, since s(l/2) = l/1r, or

1/4, 7r

(

1

2n

l

1

= 1rs(l/4) from (1.19), or

(Leibniz' series).

We extend our investigations to include the notion of an angle. In geometry, given two half-lines OA and OB meeting at a point 0, the angular sector AOB is the convex region between OA and OB. The angular measure (or simply the angle) between the half-lines can be defined by comparing this sector with a standard one-for instance that contained between two lines at right angles. Equivalently, and more conveniently from our point of view, we can define the angle as the length of the arc of a unit circle, centre at O, which is cut off by the half-lines OA and OB. If we introduce coordinates with O at the origin and take A and B at unit distance from O then we can take A at (1, 0). The properties of s and c established in section 1.7 show that B = (c(t 0 ), 1rs(t 0 )) for a unique t 0 E (0, 1), and that t --+ ( c(t), 1rs(t)) maps (0, to] bijectively to the arc AB. The length of this arc, which we have defined as the angle between OA and OB, is equal to

!,'' y'1r4 s 2 (i)

+ 1r 2 c2 (i) di= ,r !,''di= 1rio.

Let P be the point ( c( t 0 ), 0). The geometrical definitions of sine and cosine by ratios of sides of a right-angled triangle show that

OP

.

PB

cos(1rto) := OB = c(t 0 ) and sm(1rto) := OB = 1rs(to),

(1.30)

for O ::; t 0 ::; 1, i.e. for positive angles of at most two right angles. But for other angles we can appeal to the symmetry and periodicity of s, c established in section 1.3, and of sin, cos defined geometrically, to show that in fact (1.30)

CIRCULAR FUNCTIONS

33

Figure 4

Sine and cosine

is valid for all real t. But sand care entire and so we can use (1.30) to extend the geometric circular functions sine and cosine to C as entire functions, with the properties already established. We can then introduce the other circular and exponential functions, and the results of the earlier sections show that they have their expected properties. For convenience we make a list of the principal results and definitions in the conventional notation which we shall use from now on, beginning with the extended version of (1.30) above: cos(1rz) = c(z),

sin(?rZ) = 1rs(z) for z EC.

In particular sin is increasing from [-1r /2, 1r /2] onto [-1, 1], and cos 1s decreasing from [O, 1r] onto [-1, 1]. Fort we have

t(z) = s(z)/c(z), so tan(?rZ) := sin(?rZ)/ cos(?rZ) = 1rt(z), and in particular tan is increasing from (-1r /2, 7r /2) onto R. For the other elementary functions we find

c(z) s(z)

7r

( ) =1rcot(1rz), tan 1rz

ELLIPTIC FUNCTIONS

34

G1 (z)

1

=

s(z)

= 1r csc( 7rZ).

The exponential function is defined by exp(z) := e(z/21ri) = I:~ zn /n!, and we write as usual ez for exp( z), where the addition formula for e shows that ek is the k th power of e := I:~ 1/n!, at least for k E Z. More generally we can define arbitrary powers by

which makes sense for all x E C and a E established show that ( abt

c-. Properties of log and exp already

= axbx

when ~a 2: 0, ~b 2: 0 and at least one of these inequalities is strict ( and is so in particular for a, b positive), and that

for all x, y E C. For a EC, the complex power (1 + a/zY = exp(z(log(l + a/z)) is defined for lzl > lal. Then if we write f ( x) = log(l + ax) and h = 1/ z we find zlog(l + a/z) = log(l + ah)/h-+ f'(O) = a ash-+ 0. Hence for all a EC, limz-+oo(l+a/zY exists and is equal to ea. Ifwe combine the power series for exp and log we get the slightly stronger result that

(1

+ a/ z Y = ea(l + 0(1/lzl))

as lzl

-+

oo

(1.31)

which will be needed in the next chapter. (Here J(x) = O(g(x)) as x -+ a means that J( x) / g( x) is bounded in a neighbourhood of a, and y = l +O(g( x)) means that there is a function f with f(x) = O(g(x)) and y = l + f(x).) The hyperbolic functions are defined by

= ch(z/1r), sinh(z) := sin(iz)/i = 1rsh(z/1r), and = 1rth(z/1r). trigonometric functions we have y = s( x) = sin( 1rx) / 1r if

cosh(z)

·-

cos(iz)

tanh(z)

·-

sinh(z)/ cosh(z)

For the inverse and only if x = I(y), and so sin- 1 (y) = ,r:J(y/,r) =

f J[o,y]

h

1 - t2

for y EC\ ((-oo,-1] U [1,oo)).

Similarly y = t(x) = l/1rtan(1rx) if and only if x = J(y), and tan- 1 (y) = 1rJ(y/1r) =

f

J[o,y]

~ for 1+t

y EC\ ((-ioo,-i] U [i,ioo)).

CIRCULAR FUNCTIONS

35

Naturally not only the definitions but also all the formulae of sections 1-8 can be restated in the familiar notation. For instance the formula (1.14) of section 1.2 can be written CO

t( X

cot(x)cot(y)-1 + y ) = ---'-----'---, cot(x) + cot(y)

and the formula (1.22) for quotients of sines takes the form

IT (i + _x_) = sin~(x + y). n+ sm1ry y

n,e

(1.32)

Finally we establish the duplication formula for the sine in the new notation. Since for j 0 to n - 1 the powers e- 21rij/n are distinct, they are the n th roots of unity and we can factorise xn - 1 as TI;::t(x - e- 21rij/n). Now put x e2i 0 to obtain

=

=

n-1

IT (e2i0 _ e-21rij/n),

e2in0 _ l

j=O n-1

IT (eie-1rij/n(2i)sin(0 + 1rj/n))

eine (2i) sin( n0)

j=O n-1

ein0-1ri(n-l)/2(2ir

IT sin(0 + 1rj/n),

j=O n-1

sin(n0)

=

IT sin(0 + 1rj/n).

2n-l

j=O

In particular if we divide both sides by sin 0 and take a limit as 0 get the interesting special case

-+

0 we

n-1

IT sin(1rj/n) = n/2n-l_

(1.33)

j=l

1.10

Asymptotic Estimates

In this section we find some important estimates for the asymptotic behaviour of trigonometric functions of z = x +iy for large values of y. These are derived from the series for the exponential function ex I:~ xn /n!. For positive x we compare the sum with its general term to show that ex/ xn -+ oo as x -+ +oo. For complex x the fact that lex I = e~x shows that as ~x -+ oo, lex/ xn I -+ oo

=

ELLIPTIC FUNCTIONS

36

and lxne-xl-+ 0. For logarithms, the corresponding results are that for n 2:: 1, lzn log(z)I -+ 0 as lzl -+ 0 (with z E c- of course), and lz-n log(z)I -+ 0 as

lzl-+oo. For sine we find sin(x+iy)

(eix-y _ e-ix+Y)/(2i) -e-ix+Y(l + O(e- 2Y))/(2i) { eix-y(l + O(e 2Y))/(2i)

as y-+ +oo as y-+ -oo,

(1.34)

and a similar result is found for cosine. For the tangent we have from (1.34) that tan(x

. sin(x + iy) + 0( e-Y) + zy) = - - - = { i-z+O(eY) . cos(x+iy)

as y-,, +oo as y-+ -oo,

(1.35)

and similarly for cotangent. It follows for instance that although the series L~oo tan(z + nr) with S'r i= 0 is divergent since its terms do not tend to zero, the Eisenstein convention produces the series tanz+ I:~(tan(z+nr)+ tan(z - nr)) which is geometrically convergent. By contrast, the estimate (1.34) for sine shows that 1/ sin(x + iy) -+ 0 as IYI -+ oo sufficiently rapidly for the series L~oo 1/ sin(z + nr) to be geometrically convergent; the Eisenstein convention is unnecessary here. Notice especially the striking difference in behaviour as IYI -+ oo of the series E1(x + iy) = L n,e 1/(n + x + iy) = 1rcot(1r(x + iy)) which tends to =fi7r as y-+ ±oo, and G1(x + iy) = Ln(-lr /(n + X + iy) = 7r csc(1r(x + iy)) which tends rapidly to O as y -+ ±oo.

1.11

Some Alternatives

Eisenstein indicates several alternative ways of deriving the basic relations of section 1.2, and we consider two of them here. For instance the addition formula (1.14) may be written in the form

2E1(x + y)(E1(x) + E1(Y))

= (E1(x) + E1(Y)) E2(x) which is equivalent to (1.14) if one assumes E2 = Er+ 1r 2 . 2

-

E2(Y),

(1.36)

A direct proof of this may be given using nothing more than an elaborate repetition of the identity

1 pq

1

(1

1)

= p+ q p+ q

in place of the identity for 1/(pq) 2 which we used in section 1.2. Suppose then that x, y and x + y belong to C \ Z. Then 1

(x+m)(y+n-m)

1

+

(x+m-n)(y-m)

CIRCULAR FUNCTIONS

37 1

(

x+y+n 1

(

x+y-n

1- + - 1 -) + x+m y+n-m 1

x+m-n

+

1

)

y-m

( 1.37)

and similarly

1

1

------+------(x+m)(x+m-n) (y-m)(y+n-m)

= { ¼( x+~-n 1

(x+ m)2

x~m

+ y~m

- y+Lm)

+ (y-~)2

if n -:P 0, if n = 0.

If we add these equations together then the left sides combine to give

(x

+ m)(y - m)(x + m -

n)(y + n - m) ·

(1.38)

We now sum both sides, using the Eisenstein convention with respect tom for fixed n, then with respect to n. On the left side we have

and (1.38) shows that this is in fact absolutely convergent so we can reverse the order of summation and obtain (E1(x) + E1(y)) 2 . From the right side of (1.37) we get 2E1(x + y)(E1(x) + E1(Y)), while from the right side of ( 1.38) we get

E2(x) Hence (E1(x) claimed.

+ E2(y).

+ E1(Y)) 2

= 2E1(x + y)(E1(x) + E1(Y)) + E2(x) + E2(Y)

as

38

ELLIPTIC FUNCTIONS

Notice that if we proceed as in section 1.2 and expand both sides of (1.36) in powers of y, we obtain

2(E1(x)-E2(x)y+···) G+E,(x)+···) =

G+

E1(x)--y2y+ · ·

Y-

E,(x)- (:2 +-y,+ ···),

=

and on equating constant terms we get E2 ( X) = Er (X) + 312 Er (X) + 1r 2' as expected. Combining this with (1.36) gives E 1 (x + y)(E1 (x) + E 1 (y)) E1(x)E1(Y) - 1r 2 as before. Eisenstein also hints at a possible direct proof of E 2 Er + 1r 2 , though without giving any detailed indication of a method. We suggest one possibility, as follows. Since E1(x) L~ooel/(x - k) and E2(x) = L~oo 1/(x - k) 2 the result comes down to proving that

=

=

=

l

n

9n(x)

(

2

~xn

)

k k-1

L .L

k=l-nJ=-n

l

n

2

-

~

1

k-1

1

1 )

x - j - x - k

tends to -1r 2 as n--+ oo. If we put hn rewritten as

= 1 + 1/2 + · · ·+ 1/n then this may be

In particular we have

9n(O)

2

(x - k) 2 = k ~ n j ~ n (x -j)(x - k)

(

j - k

n

n

= -4 ~ J=l

(

n+j

~

r=n-J+l

1) 1

;

-:-, J

CIRCULAR FUNCTIONS

39

and we shall show first that gn(O) - -1r 2 and then that gn(x) - gn(O) for all x E C \ Z. First notice that gn(O) may be written

=-

4 n (

n+j

~

n2 ~ j=l

~ ~

r=n-j+l

!!:.r

)

~ J

=-

n

n+j

j=l

r=n-j+l

= n 24 ~ ~

~ ~

O

f(j/n,r/n)

where f(x, y) := l/(xy). A simple continuity argument shows that this converges to the integral _

4

[1 J,l+x

lo

1-x

dy dx Y

-4

X

[1 log ( ~ )

lo

l -

X

dx X

-8 j\1 + x 2 /3 + x 4 /5 + · · ·) dx -8(1

+ 1/3 2 + 1/52 + .. ·),

and since L~ 1/(2n-1) 2 = L~ 1/n 2 -I:~ 1/(2n) 2 = (1-1/4)1r 2 /6 = 1r 2 /8, this is equal to -1r 2 as required. It remains to show that gn ( x) - gn ( 0) - 0, for which we have the estimate

We split the summation into two parts, for 1 ~ j ~ n/2 and n/2 < j ~ n. In the first the sum over r is dominated by 4j / n and so the sum over j is ~ 4lxl 2 (4/n) L~ 1/lx2 - Pl - 0. In the second the sum over r is dominated by 2j and so the sum over j is::; 8lxl 2 I:r:; 2 1/lx2 - j 2 I--+ 0 as required.

1.12

Exercises for Chapter 1

1.1 Show that

I:7;;;{ Ek(j/n) =

0 if k is odd, and find the value of

I:7;;;{ E2(j/n). 1.2 Use the dominated convergence theorem to show that (1 + z/nt - ez as n - oo for all z E C. 1.3 Use the dominated convergence theorem to show that f 0n(l t/nrtx-ldt-,. fa°° e-ttx-ldt as n-,. 00 for all X with ?Jrx > 0. 1.4 Use the infinite product for s to show that s(nx) and find the constant J{ explicitly.

= K rr;- 1 s(x + r/n)

ELLIPTIC FUNCTIONS

40

1.5 Prove the addition formula for E 2(x)

= 1r 2 cosec 2 ( 1rx) in the forms

(E1(x) + E1(Y)) 2 ' 2 _E ( )-E ( )+ (E3(y)-E3(x)) 2 2 1r x y E2(Y) - E2(x)

2

Notice that the functions E 1 , E 3 are algebraic functions of E 2 by (1.12) and (1.11). The second of these results can be written in the alternative form

£(x + y) + £(x)

+ £(y) = (

£'(y) - £'(x)) 2 £(y) _ £(x)

where £(x) := E 2(x)-1r 2/3, which is an obvious precursor (and limiting case) of the addition formula (3.37) for the Weierstrass elliptic function p.

1.6 For lzl < 1 we have ?R(l + z) > 0 and hence from the discussion in section 1.9, the power (1 + z)a is defined for all a E C. Also for n EN the binomial coefficient (~) = a(a - 1) ···(a - n + 1)/n! is well-defined for all a EC. Prove that the binomial series I:~ (~) zn converges absolutely for all complex a and lz I < 1, and that its sum is ( 1 + z )a. 1. 7 Show that for sufficiently small r, the argument of increasing function oft.

I:~ an ( reitt

is an

1.8 Show that if f is holomorphic and non-zero on a star-shaped region G then f has a holomorphic logarithm on G. (Hint: consider g = ~zo,z] f' / f .) With the same hypotheses on f, deduce that it has a holomorphic square root on G. 1.9 This exercise uses the exponential function to introduce the concept of the winding number of a closed curve in C about a point. The exercise is not used until chapter 6. Let, be a curve in C, i.e. a continuous piecewise continuously differentiable function 1 : [a, b] _,. C where [a, b] C R. Let c be any point not in the range of 1 . Define 1 . 21r'l

Wnd(,,c) := -

lb ,'( a

I

( ) U

u) -

1 . 27r'l

du= C

1

dz

--.

"Y Z - C

(It is usual to restrict the definition to closed curves because of (ii) and (iii) below, but the definition makes sense in general.) Then we have the following properties of W nd: ~ (i) For a ::; t ::; b let F(t) = fat "Y(u)-c du so that F( a) = 0 and F(b) = 21riWnd(,, c). Then G(t) := e-F(t)(,(t) - c) is constant on [a, b]. (Proof: show that G'(t) = 0 on the intervals where,' is continuous.)

CIRCULAR FUNCTIONS

41

(ii) For closed curves Wnd(,,c) is an integer. (Immediate from (i) if

,(a)= ,(b).) (iii) For closed curves, W nd( 1 , c) is constant as a function of c on the connected components of the complement of 1 , and in particular is zero on the unbounded component. (Follows from (ii) and property (ii) of connected sets.) (iv) If f is analytic at zero, say f(z) = ao + anzn with n 0 2:: 1 and an 0 -=fa O then for small enough r, Wnd(f(C(O, r)), ao) = no. (Write f(z) - ao zn°g(z) etc.) (v) Let f be analytic at zo and not constant. Then the range off contains a neighbourhood of f(zo). (The open mapping principle for analytic functions.) (vi) Let P be a polynomial of degree N 2:: 1, i.e. P(z) = anzn, aN -=fa 0. Show that for sufficiently larger, Wnd(P(C(O,r)),O) = N. If ao = 0 then P(O) = 0, otherwise if a0 -=fa O then for sufficiently small r > 0, Wnd(P(C(O, r)), 0) = 0. Deduce that for some z E C, P(z) = 0. (The fundamental theorem of algebra.)

I::

=

I::

2

Gamma and Related Functions We continue our study of the series Ln(x + n)-k from chapter 1, but now we vary the definition by summing only over the non-negative integers, giving Hk(x) = I:r(x + n)-k. This results in a sequence of holomorphic functions with some properties analogous to the circular functions, but without the symmetric property of being even or odd. The corresponding infinite product p(x) x f1~(1 + x/n)e-xfn leads to the Gamma function and its related definite integrals in sections 2.3 and 2.4. A new feature is the appearance of interesting asymptotic formulae which are derived in section 2.5. The last part of the chapter introduces the Fourier transform and briefly establishes its most important properties. In particular we find the Fourier transforms of r(a + ix) and cosh-n ( x) which both illustrate the theory and lead to important applications in chapter 8.

=

2.1

Euler Series

We define the series

which, for k 2: 2 at least, defines a sequence of continuous functions on C* := C \ {0,-1,-2, ... }. Termwise differentiation shows that the sequence (Hk) satisfies the same recurrence as (Ek), namely

H~(x)

=

= -kHk+i(x) for

x

E C*,k

2: 2.

(2.1)

For k l the difficulties are different from those in chapter 1 since no rearrangement or grouping will make 1/(x + n) convergent. Instead we use (2.1) for motivation to find a function H1 on C* which satisfies

I::r

44

ELLIPTIC FUNCTIONS

This suggests that we should integrate the first term -1/ x 2 by replacing it by its antiderivative 1/x, while the other terms are integrated from Oto x. This leads to

1 L -+ 00

H1(x) :=

X

n=l

1 I:--, 1 - +1 n - -n1) = --x n( + n) 00

(

X

n=l

X

(2.2)

X

where the series is absolutely convergent on C*. Note that H1 has a simple pole with residue 1 at each integer n :S 0. Replacing x by x + y and expanding in powers of y gives the expansion 1 ----n1) x +n +y

00

H1(x+y)

1 = --+ L

x

+y

(

n=l

!+ ~ (-yl L.-J xk+l

X

k=l 00

1

(

1

(-y l

00

)

+"' ---+"'-~ x+n n ~(x+n)k+ 1 )k H,(x) + ; ~ (x ;~)k+l 00

00

(

00

H1(x)+ L(-y)kHk+1(x), k=l

which is valid for IYI < dist( x, {O, -1, -2, ... }), the inversion of the order being justified as in section 1.1. In particular H 1 is holomorphic on C* and satisfies from which it follows that each Hk is holomorphic on C*, and (2.1) is reestablished, this time for all k ~ l. The power series for H1, valid for O < lxl < 1, is obtained from (2.2); _!_

x

+ ~ (~ (-_x)i L.-J

00

00 (

j=l

1

1

n

j=O

n=l

1 ;+L

_ ]:_)

L.-J nJ+ 1

1)

LnHl n=l

00 .

- + 2 L'YH1(-x)1' X

j=l

(-x)i

GAMMA FUNCTIONS

45

where the double series is absolutely convergent and, by contrast with ( 1.6), all sums 'Yk = 2 I:~ l/nk are involved, instead of only those in which k is even. Repeated differentiation gives for k 2 2 and O < !xi < 1,

Hk(x)

1

~ = xk + !--,

(j1) · · k- l (-1)1,HixJ- . k

J=k

In particular, for k 2 2, H k has a pole of order k at each integer n ::; 0 with residue 0. In place of the periodicity of the functions Ek, we have

Hk(x

1 k fork 2 1,

+ 1) = Hk(x) -

X

and in particular since H 1 (1) = 0 we have successively H 1 (2) = -1, H1(3) = -1 - 1/2, and generally H1(n + 1) = -1 - 1/2 - · · · - 1/n = -hn, where hn stands for the 'harmonic number' 1 + 1/2 + · · · + 1/n. The functions Ek and H k are related in the following way. For k = 1 we see that 00

1 1 ; - l-x

+ n=l L

(

1 1) x+n - l+n-x

00

1 1 -+--+ x x-1

1 1) ~ --+ x+n x-n-1 ' (

which is the Eisenstein series for E 1 with the terms in 1/(x - n) shifted by one place which clearly has no effect on the sum. Hence

and it follows by repeated differentiation that

Hk(x)

+ (-ll Hk(l -

x)

= Ek(x) fork 2 2.

For k 2 2 the duplication formulae n-l

nk Hk(nx)

= L Hk(x + j/n) j=O

follow by direct rearrangement since the series are absolutely convergent. For k = 1 we have Hf= -H2 and so nH1(nx) =A+ I: H1(x + j/n), where the constant of integration has to be determined. We have

1;t

n-l

A= (nH1(nx) - H1(x)) -

L H1(x + j/n)

j=l

ELLIPTIC FUNCTIONS

46 and so letting

x ---+

0 we get n-1

A

-

L H1(j/n)

j=l

-f (~ + f (~ - ¼)) + j=l

-

lim

m--+oo

J

~ (~ + ~ (-n.- .!.) ) 6 J L..i kn + J k j=l

k=l

n-1 -

-

J~oo

J

k=l

(

n

1

n-1 m

L -:J + L L

(

kn: · J

j=O k=l

j=l

1))

k

lim (nhnm+n-1 - nhm)

m--+oo

lim (n(log(nm + n - l)

m--+oo

+ ,) -

n(log m

+ ,))

-nlogn, where we have used the result (to be proved in the next section) that there is a real number I such that hm - log m ---+ 1 as m ---+ oo. This gives us the duplication formula for H 1 , n-1

LH1(x+j/n)

= n(logn+H1(nx)),

j=O

and in particular n-1

L H1(j/n) = nlogn.

j=l

2.2

The Gamma Function as an Infinite Product

The series (2.2) can be integrated termwise to give the series

which is absolutely convergent on c-, since for large enough n the power series for log shows that the terms are dominated by a multiple of 1/n 2 . This suggests that we should consider the product

p(x)

:=

x

ft (( + =) 1

1

n

e-x/n)

(2.3)

47

GAMMA FUNCTIONS

whose terms are again of the form 1 + 0(1/n 2 ). Hence from section 0.5, (2.3) defines an entire function with zeros only at 0, -1, -2, ... and logarithmic differentiation shows that

p'(x) p( X)

Since p'(x + 1)/p(x + we deduce that

= H l ( X). 1) = H1(x + 1) = H1(x) p(x + 1)

1/x

= p'(x)/p(x) -

= p(l)p(x)/x,

1/x, (2.4)

which raises the interesting question of evaluating p(l)

= IT~ (1 +l/n )e-lfn =

=

Iimn-+oo(n+l)e-hn_ Since log(n+l)-logn J:+l dx/x is between 1/(n+l) and 1/n, we deduce that Sn := hn -log n is decreasing, while tn := hn-l -log n is increasing and so for all n, 0 = t1 < tn < Sn < s1 = 1, with Sn - in = 1/n _,. 0. Hence Sn and tn have a common limit called Euler's constant and denoted by 'Y, and we have proved that hn -Iogn _,. 'Y as n _,. oo. Since hn Jt+l dx/[x] and log n = ft dx / x we can write explicitly

=

I

=[

( [!] - ½)

dx.

The value of 'Y can be approximated by choosing a value for n; for instance = 0.49 ... < 'Y < s 6 = 0.65 .... A more accurate value is 'Y = 0.5772156649 ... (see also exercise 2.8); it is a notorious unsolved problem to determine whether 'Y is rational or not. With this notation p(l) = e--Y = 0.5614594836 .... The gamma function is defined using the reciprocal of p:

t6

e-'YX

r(x) := p(x)

Or

l

r(x) = e"fXX

l; oo

( (

X)

)

1 +;:; e-x/n ,

(2.5)

so that r is holomorphic on C* = C \ {O, 1, 2, ... }, is never zero, and satisfies r(1) = 1. The recurrence relation (2.4) becomes

r(x + 1)

=

e--y(x+l) p(x + 1)

=

xe--y(x+l) p(l)p(x)

=

and in particular for integer n 2:. 1, r(n) (n - l)!f(l) (n - l)!. An alternative way of writing (2.5) is

=

1 r(x)

=

xe -yx 1·1m rrn n-+oo j=l

= xr(x),

(n - l)f(n - 1)

(j + -eX -x/j) --.

J

48

ELLIPTIC FUNCTIONS lim x(x

e'Yx

+ 1) ... (x + n) e-hnx

n! x(x + 1) ... (x + n) -xlogn n! e

n-+oo

r

n.2..1!

lim x(x

+ 1).; · (x + n) n-x

(2.6)

n.

n-+00

for x E C, using the definition of, and of the general power nx from section 1.9.

Combining (2.4) with (2.3) we find

p(x )p(l - x) = p(x)(~:i(-x) = p(l)x

ft (1 -

,,2

/n') = p(l) sin~x)'

1

or in terms of the gamma function, e--Y

7r

r(x)f(l-x)= pxpl-x () ( ) In particular f 2 (1/2) result that

. (

Sln 7rX

) for

X (/:_

(2.7)

Z.

= 1r and since r( x) > 0 for x > 0 we have r(l/2)

the striking

= fo.

(2.8)

This also follows from (2.6) since

1/2(1/2 + 1) · · · (1/2 + n) _ ; . 11m - - - - - - - - - n 1 2 n!

1

f(l/2)

n-+oo

.

1-3···(2n+l) _ 1 ; 2 2 · 4 · · · (2n)

hm - - - - - n n-+oo

. 1-3···(2n-1) _ ; = n-+oo hm - - - - - n 1 2 2 · 4 · · · (2n - 2)

and so

_1_ r2(1/2)

=

lim _(1_·_3)_(3_-_5_)·_·_·(_(2_n_-_1_)_·(_2_n_+_l_)) 2n (2 · 4 · · · (2n)) 2 n

n-+oo

= _!_ 7r

from Wallis' product. An interesting generalisation of (2.8) is the formula

P

=

g

n-1

f(k/n)

=

(21r)(n-1)/2

fo

which is proved by writing 2

P =

n-1

IT f(k/n)f((n -

k=l

n-1

k)/n) =

7r

JI sm. (k k=l

/ ) = 1r n

(21rr-1

n

GAMMA FUNCTIONS

49

using (1.33). The duplication formulae n-1

II r(x + k/n) = (21r)Cn-1)12n112-nxr(nx)

(2.9)

k=O

can now be deduced since f' /f

= --y - p' /p = --y -

(2.2) for H 1 can be rewritten in terms of r as

~f'(x-+j/n) 0

j=O

f(x

+ j/n)

=n

H 1 , and so the formula

( - 1ogn + f'(nx)) f(nx)

or on integration n-1

II f(x + j/n) = Bn-nxr(nx),

j=O

where the constant B is to be determined. To this end we divide both sides by f(x), let x-+ 0 and get n-1

II f(j/n) = B/n.

j=l

Since the product on the left has just been shown to equal (21r)Cn-l)/ 2 /fa we have B = Jn(21rr- 1 and (2.9) is proved. For the logarithmic derivative of r the notation '1/J is customary, so that from (2.5) we have '1/J(x) := -f

00

1 (

X) = --y - H1(x) = --y -

r(x)

1 ) , L -n1 - -n+x (

(2.10)

l

and for the derivatives of '1/J (the so-called polygamma functions)

2.3

The Gamma Function as an Integral

ft

The integral e-ttx-l dt was studied in relation to the gamma function by Euler. We start with

50

ELLIPTIC FUNCTIONS

where n is a positive integer, and ?Rx > 0 to ensure convergence of the integral near t = 0. If we put t = nu and integrate repeatedly by parts we find nx

In(x)

r1 (1 -

lo

urux-l du= nx !2.

x lo

1) f\1 -

nx n(n -

r1 (1- ur- 1ux du

ur-2ux+l du

x(x+l)lo

{l ux+n-1 du

n!

nx

x(x

+ 1) · · · (x + n -

l) lo

n!

X

n x(x+l)···(x+n-l)(x+n)'

and so from (2.6), In(x)-+ f(x) as n-+ oo. On the other hand it is easy to check that for n 2: t the function ( 1 - t / n is increasing as n-+ oo, since the inequality log x = dt/t ::; x - l for x 2: 1 shows that the derivative of f(x) (1 -t/xl is positive. The limit of In(x) was shown in exercise 1.3 to be fa°° e-ttx-l dt. Hence we have shown that for ?Rx> 0,

t

ft

=

(2.11) The integral on the right here is sometimes referred to as Euler's integral of the second kind. (Euler's integral of the first kind is the integral B( x, y) defined in the next section.) We divide the range of integration in (2.11) into [O, 1) and [1, oo). The integral e-ttx-l dt defines an entire function, while in the integral over [O, 1) we can use the power series for e-t and the dominated 1 1 convergence theorem to show that e-ttx-l dt = ~r(l/n!) tn+x-l dt = 1/(n!(n + x)) and so

Ji°°

fo

~r

00

r(x)=

J:

1

fo

1

oo

1 e-ttx- dt+I: 0

'( n. n

+ x )'

which shows the behaviour of r near the points 0, -1, -2, .... 2 By putting t = u 2 in (2.11) we deduce that f( x) = 2 fa°° e-u u 2x-l du and in particular the value of the celebrated integral

{oo e-u2 du = f(l/2) = J"ir or Joo lo

2

2

e-1ru2 du

= l.

-oo

In the integral which defines the gamma function we may consider introducing a complex parameter into the exponential to get

GAMMA FUNCTIONS

51

here however we may not use the brute-force substitution u = ct without the use of the Cauchy integral theorem. Instead we can proceed by noting that differentiation under the integral sign as in section 0.6 shows that f is a holomorphic function for ~c > 0 and that its derivative there is given by

so cf'(c) + xf(c) = 0 for ~c > 0. It follows that for ~c ex f(c) = constant = f(l) = f(x), or

f(c)

=

1"'

e-'ttx-l

dt

= c-'r(x) for ~c > 0,

putting x = 1/2 and substituting t 2 for t gives ~c > 0. We can extend this a little by defining

g(y) :=

1:

~x

> 0 we have

> O; 2

f~ e-ct dt 00

(2.12)

Mc for

e-,(t+iy)' di

for real y, when differentiation with respect to y gives g'(y)

1: 1: =

-2ic(t

+ iy)e-,(t+iy)' di=

[;e-,(t+iy)'[

= 0,

00

and so g(y) = constant = g(O) = e-ct'+ 2 frtydt

Mc, or

= e-,y'

f

for y ER, ~c > 0,

(2.13)

which will be useful in section 2.6. Writing c = a+ ib in (2.12) gives

1"'

e-at ( cos(bt)

- i sin(bi) )tx-l di

= (a + ib)-'r( x) for a > 0, ~x > 0.

(2.14) We consider next what happens to (2.14) if we let a -+ O+. The integrals cos(bt)tx-l dt and sin(bt)tx-l dt are convergent for O < ~x < l, b E R \ {0} as we can see by integration by parts to show convergence at infinity. Also from exercise 0.6, these integrals are the limits as a --+ O+ of

ft

ft

ELLIPTIC FUNCTIONS

52

the corresponding integrals containing a factor 0 < ~x < l we have

1

00

e-at.

Hence for b > 0 and

= e-i1rx/ 2 b-xr(x).

(cos(bt) - isin(bt))tx-l dt

In particular for x E (0, 1) we can separate real and imaginary parts to get

1"' cos(bt)t"- dt 1"' sin( bt )tx-l dt 1

cos( 7rX/2)b-xr(x ), sin( 7rX /2)b-xf( X).

We can give this a more general appearance by putting tY, (y > 0), in place oft so that

1"' cos(bt")t•Y-l dt

(l/y) cos(7rX/2)b-xf(x),

1"' sin(btY)txy-l dt

( 1/ y) sin( 7rX /2)b-Xf ( x),

for y > 0 and O < x < l. These formulae contain a number of striking special cases. For instance if b = l, y = 2, x = 1/2, then

1"' cos(t

2

)

dt =

1"' sin(t

2

)

dt = .,fi/2,/'i,

the Fresnel integrals, or more generally for O < x

1"' 1"'

cos(u/2)I'(x + 1),

cos(t*) dt sin(tlfx) dt

l then the integrand no longer tends to zero although the integral remains convergent. For instance with y = 6, x = 2/3 we have

1"' 2 .4

cos(t 6 )t 3 dt

1"'

= 3I'(2/3),

6

sin(t )t 3 dt

= 3v'3r(2/3).

Beta Integrals

In place of the integral In( x) which was introduced section 2.3, we consider B(x,

y)

:=

1'

(1- t)"- 1 tY- 1 di,

GAMMA FUNCTIONS

53

where x, y are required to have a positive real part to ensure convergence. The substitution t ---+ l - t shows that B(x, y) = B(y, x). For non-integer values of y, repeated integration by parts as in section 2.3 does not help. Instead we derive a relation between B(x, y) and B(x + 1, y) as follows. Firstly we have for ~x > 1,

1' 1'

B(x,y)

(1 - t)(l - t)'- 2 t•- 1 dt (1 - t)'- 21•- 1 dt -

1'

(1 - t)"- 2tY dt

B(x - 1, y) - B(x - 1, y + 1).

> 1,

Integration by parts, again with ~x 1

x-11 +-

1 B(x, y) = -tY(l - tyi:- 1 1 y

Y

O

1

gives

x-1 (1 - t)x- 2 tY dt = --B(x - 1, y + 1). Y

O

Combining (2.15) and (2.16) gives B(x, y) putting x + l for x,

B(x, y)

(2.15)

= B(x -

x+y = --B(x + 1, y) for X

~x

(2.16) 1, y) - 6B(x, y), or on

> 0, ~y > 0.

(2.17)

We can apply (2.17) repeatedly to get

x+y --B(x + 1, y)

B(x, y)

X

(x+y)(x+y+l)B( + 2 ) x(x+l) x ,Y (x+y)(x+y+l)···(x+y+n)B( ) x+n+ 1 ,Y. ( ) ··· (x+n ) xx+l But as n

---+

oo,

nY B( x + n + l, y) which tends to Thus we have

B(x, y)

= n•

1'

(1 - t)'+nty-l dt

fa°° e-tty-l dt = I'(y)

=

= 1\1 - t/n)'+nty-l dt

as n---+ oo (exercise 1.3), since ~y

> 0.

+ y) (X + y + 1) ... (X + y + n) n -y) f ( y) x(x + 1) · · · (x + n) ( ( X + Y) (X + y + 1) · · · ( X + y + n) n !n x ) f ( y) n!nx+y x(x + 1) · · · (x + n)

lim ( ( X n-+oo

Jim n-+oo

I'(x)I'(y) I'(x+y)'

(2.18)

54

ELLIPTIC FUNCTIONS

using (2.6). This result gives the holomorphic extension of B(x, y) to all x, y EC\ {O, -1, -2, ... }. The substitution t sin 2 0, 0 ~ 0 ~ 1r /2, gives

=

B(x, y)

= 2 lor12 sin 2 x-l 0 cos 2y-l 0 d0 for ~x > 0, ~y > 0.

(2.19)

The reader equipped with the formula for change of variables in a multiple integral may choose to derive (2.18) differently by writing f(x

+ y)B(x, y)

21

e-r r•+y-l dr

41 00

e-r' r 2'+ 2 y-l dr

41 1' 00

12

41001=

1=

1' 1' 12

00

sin 2x-l 0 cos2Y-l 0 d0

12

sin 2 x-l 0 cos2 Y-l 0 d0

e-r'(rsin0) 2"- 1 (rcos0) 2 Y-l d0rdr

e-(u'+•')u2x-lv2y-1 dudv

e-"ux-l du

1=

e-•vy-l dv

= r(x)l'(y).

Another useful expression for B(x, y) comes from the substitution t u/(1 + u), 0 ~ u < oo which gives

B(x,y)

In particular if

x

)x-1

rXJ(

lo lof

1: U

d

(1 + :)Y + l

00 (

u•-)l + du for ~x 1+ U X y

+ y = 1 we have from (2.7)

l

> 0, ~y > 0.

(2.20)

the interesting special case

oo ux-1

0

--du l+u

r(x)f(l - x)

7f

= Sln . ( ) for O < ~x < 1.(2.21) 7rX

Formulae (2.20) and (2.21) may be restated as Fourier Integrals, as we show in section 2.6.

2.5

Asymptotic Estimates

In this section we establish the famous formula of Stirling which gives the asymptotic behaviour off( x) for large values of Ix!. We derive it first in the

GAMMA FUNCTIONS

55

form of an equation ((2.25) below) from which we can find as many terms in the asymptotic expansion of f(x) as are needed. The starting point is the formula (2.6) which can be written

1

=

_1_ xf(x)

+ 1)

f(x

lim ( ( x

+ l) ( x + 2) · · · ( x + n) n n!

n--+oo

lim ( ( 1 + x) ( 1 + x / 2) · · · ( 1 + x / n) n -

n--+oo

-x)

x)

exp[}!..'! (tlog(l+ x/r)- xlogn)] .

(2.22)

Notice that in this expression the (principal) logarithms are well defined, and the formula is valid, for all x with 1 + x E c-, i.e. for x (f. (-oo, -1]. We can not simply take logs in the definition of r as an infinite product, since we have no way of knowing which value of log r should be taken. The equation (2.22) leads us to consider the behaviour of the sum I:~ log(l + x/r) for which we use a simple version of the Euler-Maclaurin summation formula. Suppose that a ( complex valued) function f is given which we suppose to have a continuous derivative on (0, oo ). Define 'lfJ(t) := t - [t] - 1/2 fort E R; equivalently 'lfJ(t) t - n - 1/2 for n '.S t < n + l, n E Z. (This is of course not the same function 'Ip which was used to denote f' /fin (2.10)). Then we can write

=

r+l

Jn

r+l

= Jn

f (t) dt

1

2

'lfJ 1 (i)f(t) di= 'lfJ(i)f(t)J~+l -

(f(n + 1) + f(n)) -

r+l

Jn

r+l

Jn

'lfJ(i)f'(t) di

'lfJ(t)f'(t) dt,

and so on summing over n,

J, l

n+l

f (t) dt

1

1

= 2J(l) + /(2) + · · · + f(n) + 2J(n + 1)

J,

-

n+l

'lfJ(t)f' (t) dt, or

1

n

J,

n+l

1

+

J, l

1

f (t) dt + - ( f ( 1) - f (n + l)) 2

n+l

'ljJ(t)f'(t) dt.

(2.23)

This is the first version of the Euler-Maclaurin summation formula which we need. For f(t) = log(l + x/t), we have f'(t) = l/(x +t)- l/t and so (2.23)

56

ELLIPTIC FUNCTIONS

gives n

r+l

L log(l + x/r)

11

log(l

+ x/t) dt

1

+ 2(log(l + x) +

J,

n+l

(

'!jJ(t) X

l

log(l

+ x/(n + 1)))

1 - -1) dt.

+t

(2.24)

t

The first integral may be evaluated explicitly to give

r+l log(l + x/t) dt

((x + t) log(x + t) - t logt)l~+l

11

(x + n + 1) log ( x + n + 1) - (n + 1) log ( n + 1) - ( X + 1) log ( X + 1) X

+ 1) + (x + n + 1) log( 1 + - -1 ) n+ (x + 1) log(x + 1).

x log( n

Put this into (2.24) to get

x

n

L log(l + x/r)

log(n + 1)

- (x

n

+ (x + + 1/2) log (1 +

+ 1/2) log(x + 1) +

J,

n+l

'!jJ(t)

1

(

_x_) n+l

X

1 - -1) dt.

+t

t

Now substitute this into (2.22) and take a limit as n --+ oo, noting that + n + 1/2) log(l + x/(n + 1))---+ x. We find

x log(n + 1) - x log n---+ 0 and (x

r(x~l) =exp[x-(x+l/2)log(x+l)+

1,= ,J,(t)c: 1 -D dtJ,

or replacing x + 1 by x we have f(x)

= ce-xxx-l/ 2 exp

[-

f 11

00

'!jJ(t)

X

+t -

It

1

dt] '

X (/-.

(-oo, O]

(2.25)

where C = exp(l + '!jJ(t)/t dt). This equation is our first form of Stirling's formula. We shall show shortly that C = ~ (so that for the integral '!jJ(t)/t dt we have the curious value -1 + ½log(21r)). The principal terms e-xxx- 1 / 2 dominate the behaviour as lxl ---+ oo as we shall show by investigating the behaviour of the integral 1P (t) / (x + t - 1) dt in (2. 25) .

It

It

GAMMA FUNCTIONS

57

Define x(t) = (t - [t] - 1/2) 2 - 1/12 so that x' (t) = 2'!f(t) on each ( n, n + l) and the constant term -1/12 is chosen so that J:+1 x(t) dt = 0. It follows that Ix( t) I ~ x(O) = 1/6 for all t E R. Then

00

1 1

11

-'lf(t) - - dt x+t-l

00

-

2

1

!

-x'(t) - - dt x+t-l

x(t)

+ ! 100

dtloo

2 x+t-l

2 1

1

x(t)

(x+t-1) 2

dt

__1_ + loo x(t) dt 2 · 12x 1 (x+t-1)

We aim to show that the integral here is bounded by a multiple of 1/lxl if lxl--+ oo in any sector of the form S0 := {x: jargxj S 1r- 8}, 8 > 0 (that is, in any closed angular sector of opening less than 21r which excludes the negative real axis). Su pp ose firstly that I arg x I S 1r / 4 so that x = u + iv where 0 S lvl S u and u 2:: lxl cos( 1r / 4) = lxl/ \/'2. Then

I~ f

(x

+x?21) dtl 2


0, If (u) I < Ae-alul then j is holomorphic for IS'vl < a/21r. However it is not generally true that lf(v)ldv < oo; an immediate 00 example is when f(u) = 1 for lul :s; 1, f(u) = 0 otherwise. Then }(v) e- 21riuvdu = sin(21rv)/(1rv) for v f:. 0, }(O) = 2 and lfl diverges. 1 00 ~ther interestinq examples of Fourier tr~nsfo~ms are 2 (1) /( u) = e-a1ru , ~a > 0 when (2.13) gives f (v) = a- 1 ! 2e-1rv I a ( another eigenfunction), and (ii) f( u) = e-1raiul, a > 0 when an elementary calculation shows that

J~

f

J~

}(v)

{

00

e-21riuv-1raudu

Jo 1f

+ J_o

e-21riuv+1raudu

-oo

( 22a 4 2) for a

+

V

IS 0, u E R.

2. 7

Exercises for Chapter 2

2.1 Find the region of validity for the formula

f(x) I'(x+h)

= (1 + h/x)e'fh

IT (1 +

n=l

_h_) e-h/n. x+n

2.2 Find the region of validity for the formula I'(a+iv)I'(a-iv)

f2(a)

= rroo n=l

(1+

v

(n

2

+ a)2

)-l

2.3 (i) For a function f satisfying the Fourier inversion formula (2.36), show that JAA( x) = f (-x) and hence that the Fourier transform (2.35) has period 4. (ii) For a function fas in (i) which is also even, show that j is even, and 00

}(y)

00

= 2 fo f(x) cos(2irxy)dx and f(x) = 2 fo }(y) cos(2irxy)dx.

GAMMA FUNCTIONS

67

In particular, the Fourier transform, restricted to even functions, has period 2.

(iii) For a function f as in (i) which is also odd, show that

1

00

f (y)

= -2i

f ( x) sin( 2,rxy )dx and f ( x)

= 2i

1

j is odd, and

00

j (y) sin( 21rxy )dx.

Thus for odd functions it is natural to modify the Fourier transform and define f := i} when

f(y)

=

21

00

f(x)sin(2,rxy)dx and f(x)

=

2['° i(y)sin(21rxy)dx.

This modified Fourier transform, restricted to odd functions, also has period

2. (iv) Since any function can be written as a sum of even and odd components, the results of (ii) and (iii) show that the domain of the Fourier transform can be expressed as a sum of two invariant subspaces, on each of which the transform (modified as appropriate) has period 2. 2.4 Show that for bounded CI-functions, (f

*gt= j · g.

2.5 Find the asymptotic behaviour off( u +iv) for v -f:. 0, u

-+

-oo.

2.6 (i) Let Vn denote the volume of the n-sphere, Sn := {(x1, ... , xn) E Rn : + · · · + ~ 1}, so that Vo= 1, Vi= 2, Vi= 1r, V3 = 41r/3. Show by considering the cross-section of Sn by the plane Xn = t, - l ~ t ~ l that

xr

x;

J_-1 l

v;

v; n-l

(1 - t2)(n-1)/2dt

n-1

= v;

2

2nr ((n + 1)/2) n-1 f(n+l)

ftr( (n + l) /2) f((n + 2)/2) ·

Hence 1r-n/ 2 r((n+2)/2)Vn is independent of n and so must equal 1, its value when n = 0 (or n = l, 2, 3 from the values give above). Thus we have for the volume of Sn, 7rn/2 7rn/2 Vn

= f((n +

2)/2) - (n/2)!.

(ii) Consider the unit cube in Rn with vertices at ( e1, ... , en) where each is O or 1. Place 2n spheres, each of radius 1/2, one at each vertex, and place one further sphere with centre at (1/2, ... , 1/2), the centre of the cube, to touch each of the other spheres externally. For which values of n does this further sphere lie entirely inside the unit cube? ej

2.7 If we change the sign in (2.30) we obtain a new sequence En(t), the Euler polynomials,

ELLIPTIC FUNCTIONS

68

These have many properties which are analogous to those of Bernoulli polynomials. Show in particular that 2n En (1/2) is an integer, which we shall denote simply En, and derive recurrence relations for En and En ( x) analogous to (1.27) and (2.31). 2.8 Use (2.33) with k

= 3 to obtain the value 'Y = 0.57721... with an error

< 10- 5 . (The estimate IBn(x)I::; n!/(1rn(2n- 1 - l)), n ~ 2, x E Rwhich is exercise 4.2, will be needed.)

2.9 For positive integers n, the double factorial is defined by f(n) = n!! := n!(n - 1)! · · · 2!1!, so that

f(l)

= 1 and f(n) = n!f(n -

1) for n 2:: 2.

Define a suitable function of a complex variable which satisfies this recurrence, and establish asymptotic formulae for it, analogous to Stirling's formula for

r. 2.10 Find the value of f 07r 12 sinn 0 d0 by induction (integrate by parts to obtain a recurrence relation), and check your result against (2.19).

3

Basic Elliptic Functions In our first chapter on circular functions we studied periodic functions with a single period; these functions had features such as zeros and singularities which were repeated in a single direction. We have now arrived at our main subject matter, the elliptic functions which, among meromorphic functions, are characterised by being doubly periodic; that is there exist complex numbers u, v with v/u (J. R such that both u and v are periods of the function. In this case the period set is a lattice of the form L = L(u, v) = {mu+nv: m, n E Z} and the behaviour of the function is repeated in the two distinct directions given by the generators u, v. Properties of lattices are discussed briefly in the first section. We shall consistently write r for the ratio v /u, and replace v by -v if necessary to ensure that S'r > 0. Examples of elliptic functions with periods u and v are given by the Eisenstein series

when k > 2. All summations are over Z unless otherwise indicated and so we can ~ite as here Ln in place of L~=-oo, and Ln,e when the Eisenstein convention is in force. We can carry out the sum over m explicitly to obtain the fundamental formula (3.2) which expresses each A as a sum of circular functions and makes it possible to use the results of chapter 1. Convergence problems are more delicate than in chapter 1 and are resolved in section 3 .2 by making use of the results of chapter 1 rather than by direct methods. We shall show that each Fk is a holomorphic function on C \ L, with power series expansions which are analogous to those of section 1.1. The transformation T ~ -l/r corresponds to the change (u, v) to (v, -u) of generators of the lattice and results in rotation formulae-for instance (3.16)-which have no analogue for circular functions. Addition formulae and the nonlinear relations between the functions Fk are established in section 3.3, building on the results of section 1.2. The differential equation (3.26) for F2 is particularly important here, and

70

ELLIPTIC FUNCTIONS

leads to the introduction of the Weierstrass functions

( = Fi + 62x in the final section of the chapter.

r =

F2

-

82 and

The limiting cases as S'r -+ oo return us directly to circular functions; these results are in the exercises. The limiting cases as 8-r -+ 0 are much deeper and are considered in chapters 7 and 8.

3.1

Lattices

As described in the introductory paragraph, a lattice in C is a set of the form

L(u, v) :={mu+ nv: m, n E Z}, S'(v/u) > O}.

More abstractly a lattice is an additive subgroup of C which is (group) isomorphic to Z 2 with generators which are independent over R. From this we make the observation that knowing the lattice as a point set (for instance as the period set of an elliptic function) does not determine the generators uniquely; for instance the pairs (v, u) or ( u, -v) or generally (m1 u + n1 v, m2u + n2v) for any m1, n1, m2, n2 with m1 n2 - m2n1 = ±1 all generate the same lattice as ( u, v). From section 3.2 onwards we shall partially standardise by dividing by u so that the lattice has the form L(l, r) where S'r > 0. A more complete standardisation would allow us to have in addition lrl ~ 1 and -1/2::; ~r::; 1/2 with some further restrictions on the boundary of the region; we postpone this until Chapter 7. We shall be particularly interested in the change of basis ( u, v) -+ ( v, -u) (which corresponds to r -+ -1/r) since this corresponds to changing the order of summation in the defining double series. More general lattice transformations are discussed in [We]. The elliptic functions defined by (3.1) have singularities at points of L and are analytic elsewhere. It follows that the period lattice (though not its generators) is uniquely determined by the function. Points x, y for which x - y E L are said to be congruent mod L. Given a pair of generators u and v, the period parallelogram is the set

P( u, v)

:= { su

+ tv : 0 ::; s < 1, 0 ::; t < 1}.

Evidently every point in C can be written uniquely in the form l + p where l E L and p E P; equivalently every point of C is congruent to a unique point of P.

BASIC ELLIPTIC FUNCTIONS

3.2

71

Basic Elliptic Functions

We begin by considering the series

1

A(x):=LL( x+m+nr )k.

(3.1)

n,e m,e

Here k 2 1 is an integer, 8r > 0, L = L(l, r) and x E C \ L. We write Fk(x Ir) for Fk(x) when we want to draw attention to the parameter T. As noted above, both summations are over Z and the Eisenstein convention may be required in one or both. The order in which the summations are performed may be significant; the way (3.1) is written indicates that the summation is first over m and then over n. We recognise that the inner sum is of the type used in section 1.1 to define the circular functions Ek so that

(3.2) n,e

This expresses the fundamental relation between circular and elliptic functions, that each elliptic function Fk is formed by the addition of copies of the corresponding circular function Ek, each copy being translated by multiples of a fixed number T whose imaginary part is non-zero. Notice that if T is purely imaginary and x is real then the conjugate of Ek(x + nr) is Ek(x - nr) and so Fk is real on the real axis but in general the values of Fk on the real axis will be non-real. To investigate the convergence of (3.1) we need to estimate the growth of the terms. For this we have the inequality (0.1)

1/(2 + lrl) < vm 2 + n 2

1

2/ I{

Ix+ m + nrl ::; vm 2

-

+ n2 '

Jl

when lxl ::; min(lml, I{ vm 2 + n 2/2) and I { = l8rl/ + lrl 2 . Hence the series ( 3 .1) will converge absolutely when the series 2 2 2 I:(m,n),t(o,o) 1/( m + n ll converges which, by comparison with the integral

f fx2+y2> 1 l/(x 2 + y 2 ?l 2 dx dy =

21r ft° rl-k dr, occurs if and only if the integer ""k 2 3. It follows that for k 2 3, the function A is well-defined by (3.1), with a sum which is independent of any rearrangement of terms. Notice that as in previous chapters, termwise differentiation shows that we have the recurrence relation F£ = -kFk+l for k 2 3. Consideration of convergence in the case k = 2 is relatively straightforward since in (3.1) the inner summation over mis absolutely convergent with sum E 2 (x + nr) 1r 2 / sin 2 (1r(x + nr)), so in this case (3.1) becomes

=

L sin (1r(x + nr)) ?T2

F2(x)

:=

n

2

.

(3.3)

72

ELLIPTIC FUNCTIONS

The estimates (1.34) show that these terms are O(e- 21rlnrl), and so the summation over n is absolutely convergent and F 2 is well-defined by (3.3). Note that this does not conflict with the failure of (3.1) to be absolutely convergent as a double series; we have simply shown that the grouping in which we sum first over m and then over n produces results which are absolutely convergent. Reversing the order of summation also produces series which are absolutely convergent but the sum is changed as we shall show later. Now consider the case k = l. The inner sum over m ( using the Eisenstein convention) is the series for E1 from section 1.1,

1 ~ ---L.t x + m + nr m,e

= E1(x + nr) = 1rcot(1r(x + nr)).

However the cotangent does not tend to zero as n -+ oo; the results of section 1.10 show that 1r cot(1r(x + nr)) = =i=i1r + O(e-lnl1r,8) as n -+ ±oo, where /3 = S'r. But these same estimates show that if we group together the terms in n and -n (the Eisenstein convention again) we obtain another absolutely convergent series. Hence F 1 is well-defined by

Fi(x)

1 LL---= L 1rcot(1r(x + nr)) x + m+nr n,e m,e

n,e

00

1rcot(1rx) + 1r L(cot(1r(x + nr)) + cot(1r(x - nr))) 00

1rcot(1rx)+1rsin(21rx)L. sm ( 7r ( x 1

1 )) . ( 1r ( x - n, ))(3.4) + nr sm

Thus we have defined a sequence of continuous functions ( Fk )'f° on C \ L for which Fk is even when k is even, and odd when k is odd. At each point of L, F 1 has a simple pole with residue 1 while fork 2: 2, Fk has a pole of order k with residue zero. It follows by inspection of (3.1) fork 2: 3, or (3.3) fork= 2, that Fk is doubly periodic for k 2'.: 2 with periods 1 and T. For F1 the situation is different, since although ( 3 .4) shows at once that F 1 ( x + l) = F 1 ( x), F 1 does not have T for a second period. Indeed

Fi(x + r) - F1(x) N

lim ~ 1r (cot(1r(x + (n + 1),)) - cot(1r(x + nr)))

N-,.oo

L.t -N

lim 1r(cot(1r(x + (N + l)r))- cot(1r(x - Nr)))

N-,.oo

-21ri, and so more generally we have for m, n E Z,

Fi(x + m + n,)

= Fi(x) -

2n1ri;

(3.5)

BASIC ELLIPTIC FUNCTIONS

73

in particular F 1 is not an elliptic function as we have defined them. [The reader acquainted with the general theory of meromorphic functions will see that a function such as Fi with a single simple pole in the period parallelogram could not possibly be doubly periodic.] Since Fi is odd, we may deduce from (3.5) the values

Fi((l + r)/2) = Fi(r/2) = -1ri

Fi(l/2) = 0,

(3.6)

which will be useful later. To find the power series expansion of F 1 about a point x E C \ L, we put x + y for x in (3.4) and expand in powers of y for IYI < dist(x, L). This gives

Fi(x+y)

=

1

LLx+y+m+nr n,e m,e

LL C+~+nr + (x+r::nr)

2

n,e m,e

(-y)i

+ ~ (x + m + nr)i+ 1 CX)

) (-y)i

+ 6~ L ~ ( )'+1 · ~ x+m+nr J CX)

Fi(x) - yF2(x)

n

m

J=2

But in the multiple sum, the terms are dominated for IYI


0. The value r = i gives (3.19)

76

ELLIPTIC FUNCTIONS

To complete this section we establish the duplication formulae. By analogy with (1.8), we consider the sum Fk(x + r/n + s,/n) which for k 2: 3 may be rearranged to give

I:;;t I:;;t

n-ln-1

LL Fk(x + r/n + s,/n) l

n-ln-1

1: ~ ( + ~ ~ 1:~

~~

x

r/ n

+ s, / n + p + q, )k

n-ln-1

n•

(nx

+ (np+ ,{ + (nq + s)r)'

nk Fk(nx).

(3.20)

By integration we deduce n-ln-1

LL F2(x + r/n + s,/n)

n-ln-1

LL Fi(x + r/n + s,/n)

=

nFi(nx) - Cx + D,

r=Os=O

where C, D may depend on n and, but not on x. Now put x of x:

+ ,/n in place

n-ln-1

LL Fi(x + r/n + (s + 1),/n)

nFi(nx + ,)

r=Os=O

- C (x

+ , / n) + D,

or

n-ln-1

LL Fi(x + r/n + s,/n) + n(-21ri) r=O s=O

-C(x + ,jn) + D, using (3.5). It follows by subtraction that C = 0 and hence that (3.20) is valid as written for all k 2: 2. To find D, put -x for x so that n-ln-1

LL F1(x -

r/n - s,/n)

=

nFi(nx) - D, or

n-ln-1

LL Fi(x + r/n + s,/n) + n(n -

r=O s=O

1)21ri

nFi(nx) - D,

77

BASIC ELLIPTIC FUNCTIONS using (3.5) again. Hence 2D

= -n( n -

l )21ri and we have

n-ln-1

LL F1(x + r/n + sr/n) = nF1(nx) -

n(n - l)1ri

r=Os=O

=

in place of (3.20) when k = l. From (3.20) with k subtracting 1/ x 2 from both sides and taking a limit as x

---+

2 we deduce, by 0, that

n-ln-1

LL'F2(r/n+sr/n)

= (n

2

-1)8 2 ,

(3.21)

and the important special case

(3.22)

It is also possible to establish what might be called 'quasi-duplication' formulae, for instance F1(x I,)+ F1(x + 1/2 I ,) = F1(2x j 2,), where there is a change not only in x but in the parameter ,-see exercise 3.6 and also (4.20). These results are related to Landen's transformation of Jacobian elliptic functions as we shall see in the exercises to chapter 6. More generally I:;.n:01 I:;,:~ F1(x + r/m + sr/n) can be related to Fi(axjbr) for suitable integers a, b; we leave the details to the reader.

3.3

Addition Formulae

To obtain addition formulae for elliptic functions our most economical route is to use the addition formulae for circular functions from section 1.2 combined with the representation (3.2). Our starting point is (1.10),

= E2(x + y)(E2(x) + E2(y)) + 2E3(x + y)(E1(x) + E1(Y)). To derive the corresponding formula for elliptic functions, put x + mr for E2(x)E2(Y)

and y + (n - m)r for y and sum over m. We find

L E2(x + mr)E2(Y + (n -

m)r)

m

E2(x + y + nr)(A(x) + F2(Y + nr)) + 2E3(x + y + nr)(F1 (x) + Fi (y + nr)) E2(x + y + nr)(F2(x) + F2(Y)) + 2E3(x + y + nr)(F1(x) + F1(Y) - 2mri) using (3.5). We then sum over n to get

F2(x)F2(Y)

=

F2(x + y)(F2(x) + F2(Y)) + 2F3(x + y)(F1(x) - 41ri

L nE3(x + y + nr) n

+ Fi(y))

x

78

ELLIPTIC FUNCTIONS

where the change of order on the left side is justified by absolute convergence. Note that all terms correspond to those of (1.10) except for the final sum S := -41ri Ln nE3(x + y + nr). Since F2(x) = Ln E2(x + nr), we have 8F2(x)/8r = - Ln 2nE3(x + nr) = S/21ri, and so

+ y)(F2(x) + F2(Y)) 0 + 2F3(x + y)(Fi(x) + F1(Y)) + 21ri :/(x + y).(3.23)

=

F2(x)F2(Y)

F2(x

Before completing the proof of the addition formula (3.30) below, we digress to deduce nonlinear relations between the elliptic functions Fk which correspond to the relations found for circular functions in section 1.2. To do this we expand (3.23) in powers of y and equate constant terms to give

(F,(x)- 2yFs(x) + 3y 2 F4(x) + · · ·)

F2(x) (:, + 62 + · · ·)

x (:,

+F,(x)+6d··)

+ 2 (F3(x) x

G+

3yF4(x)

F, ( x) - y6,

+ · · ·) + · ·)

) + 2?.8F2( r ' l - X +··· or

and so

3F4(x)

=

If we now expand (3.24) about x

F:j(x)

+ 2Fi(x)F3(x) + 21ri

0 : : (x). (3.24)

= 0 and equate constant terms we get (3.25)

We can integrate (3.24) to give

(3.26) the constant of integration being zero since all terms are odd. This is obviously the analogue of the formula E3 = E1 E2 for circular functions. Notice that oFif or = - Ln nE2(x + nr) is holomorphic for IS'xl < IS'rl. Also oFi/ or = \Ji' where 00

'11(x)

:=

L n (E1(x + nr) 1

E 1 (x - nr) - 2E1(nr)),

BASIC ELLIPTIC FUNCTIONS

79

the final term being needed to ensure convergence. Then we can integrate once more to get where 382 is found as usual by comparing expansions about x = 0. This is as near as we can get to the formula E2 = Er+ 1r 2 for circular functions. Another nonlinear relation may be obtained from (3.23) by putting x + y = z, expanding about z = 0, and equating constant terms. We obtain successively

F2(z) (F2(x) + F2(x - z))

8 + 2F3(z) (F1(x) - Fi(x - z)) + 21ri :} (z),

cl2 + 82 + .. ·)

FJ(x)+···

x (2F2(x) + 2zF3(x) + 3z 2 F4(x) + · · ·)

+(z23+

)

x (-zF2(x) - z 2F3(x) - z 3F4(x) + · · ·) .882

+ 27rZ a; + y ... ' and

F4(x)

=

2 ( . 882 F 2 (x)-282F2 x)-21rza;,

which is a second-order differential eqaution for F2 which will be considered further in the next section. If we use (3.25) to eliminate 88 2 / OT we get the alternative form

(3.27) We return to the problem of finding addition formulae. Integrate (3.23) with respect to y to get

F2(x)(F1(y)-F1(x+y))

= (F1(x)+Fi(y))F2(x+y)+21ri 8F1 OT (x+y)+¢(x), (3.28)

where ¢( x) is found from the expansion of (3.28) about y = 0. We find ¢(x) = 21ri8Fi/8T(x) = F3(x) - Fi(x)F2(x) from (3.26). (The reader has already seen sufficiently many of these arguments for us to begin omitting the details.) Put this expression for ¢(x) into (3.28) to get

F2(x)(F1(Y) - F1(x + y))

=

(F1(x) + F1(Y))F2(x + y) + F3(x + y) + F3(x) - F1(x + y)F2(x + y) - Fi(x)F2(x) (3.29)

80

ELLIPTIC FUNCTIONS

In this we put y

=z -

x and rearrange to get

(F3(z) + F3(x)) + (Fi(z) - Fi (x )) (F2(x) - F2(z )) Fi(x)

+ Fi(Y) - F3(x) - F3(y)

(3.30)

F2(x) - F2(Y)

which is the required addition formula. Note that it expresses Fi ( x + y) not only in terms of Fi but also of its derivatives F 2 and F 3 . The corresponding addition formula for F2 will be discussed in the next section on Weierstrass functions.

3.4

Weierstrass Functions

The reader may have noticed that some of our results concerning F2 , in particular (3.21), (3.22) and (3.27), are neater with F2 (x) - 82 in place of F2. This leads us to define a new function r( x) := F2(x) - 82 (the notation is due to Weierstrass) and (3.13) gives

In particular, p(x)-1/x 2 = O(x 2 ) as x---+ 0. Ifwe combine the double sums (3.1) for F2 with (3.10) for 82 then we have

r (x ) = -x12 + LL' ( (x + m1+ nr) 2 m

n

1

(m + nr) 2

)

,xEc\

L.

(3.32)

The double sum here is absolutely convergent by comparison with LL Im+ nrl- 3 and so we may rearrange the terms in any way. Formulae already established for F2 imply corresponding results for p. Without making a complete list we mention the duplication formulae

n-in-i

LL p(x + r/n + sr/n)

=

n 2 p(nx)

r=Os=O

n-in-i

LL 'p(r/n + sr/n)

0,

r=O s=O

r(l/2)

+ r((l + r)/2) + r(r/2)

0,

(3.33)

which follow from (3.20), (3.21) and (3.22) respectively. The rotation formula T-

2

p(x/rl - 1/r)

= p(xlr),

(3.34)

BASIC ELLIPTIC FUNCTIONS

81

can be deduced from (3.17) but comes more easily from (3.32) by rearrangement of terms, and the differential equation (3.35) is equivalent to (3.27). Multiply (3.35) by r' and integrate to get (r') 2/2 = 2p 3 - 3084p + const., where the constant is found from comparing expansions about x = 0 as usual. We find

(3.36) which is Weierstrass' first-order differential equation for p. The coefficients in (3.36) are often written simply as g2, g3 so that (3.36) becomes·

(r')2

= 4p 3 -

g2r - g3,

g2

= 6084,

g3

= 14086.

To deduce the addition formula for p, integrate (3.29) with respect to x to get

F1(x)F1(Y)

=

(Fi(x) + Fi(y))F1(x + y) + (F2(x +(F2(x) - F;(x))/2 + x(y)

+ y) - F{(x + y))/2

where in view of the symmetry in x, y, x(y) must be of the form ( F2 (y) Ff(y))/2+k, and the expansion about x = 0 shows that k = -382/2. We can then rearrange to give

using ( 3 .30). This is the addition formula for F 2 ; in terms of p it is

p(x + y)

1 (r'(x) - r'(y))

+ p(x) + p(y) = 4 r(x) - p(y)

2

'

(3.37)

where because of (3.36) the derivatives are algebraic functions of p(x), p(y). In terms of p, the addition formula (3.30) for F1 may be written

(3.38) which is often its most useful form. At this point it is interesting to consider the half-periods 1/2, (1 + r)/2 and r/2. Since r' is an odd periodic function we have r'(l/2) = -r'(-1/2) -r'(l/2) so p'(l/2) = 0 and similarly r'(r/2) = r'((l + r)/2) = 0. We write

=

82

ELLIPTIC FUNCTIONS

e1 := r(l/2), e2 := r((l + ,)/2) and e3 := r(,/2), where e1 + e2 + e3 = 0 from (3.33). Each of e1, e2, e3 may be written as a sum involving, similar to (3.12) for 82; these are in the exercises. We see from (3.34) with x = 1/2 that when,= i, e1 + e3 = 0 and so e2( i) = O; the value of e1 ( i) will be found in section 7.5. From e2(i) = 0 we can deduce the sum of another series like (3.19), namely 00 1 1 ; cosh 2((n + 1/2)7r) = 271" · If we factorise 4t 3 - 6084t - 14086 = 4(t - a)(t - {J)(t - ,) for a, {J, 1 E C, then a + {J + 1 = 0 since the coefficient of t 2 is zero. The differential equation (3.36) shows that each of e 1, e 2 , e3 is in the set {a,/J,,} since r' = 0 when x = 1/2,,/2, (1+,)/2. Let us assume temporarily that { e1, e2, e3} = {a, {J, 1 }, which is certainly true if e1, e2, e3 are distinct; this will be proved below and in a different way in the next chapter. Then (3.36) may be written

(3.39) We can put y

= 1/2 in (3.37) to get successively

p(x + 1/2) + r(x) + e1 r(x + 1/2)(r(x) - e1) (r(x + 1/2) - e1)(r(x) - e1)

(r(x) - e2)(r(x) - e3)/(p(x) - e1),

-(e2 + e3)p(x) + ei + e2e3, (e1 - e2)(e1 - e3),

using e1 + e2 + e3 = 0. Similarly

(p(x + (1 + ,)/2) - e2)(p(x) - e2) (p( X + 1" /2) - e3)(p( X) - e3)

= (e2 = (e3 -

e1)(e2 - e3), e1)( e3 - e2),

and

(3.40)

These formulae are analogous to (E 2 (x + 1/2) - 7r 2)(E 2(x) - 7r 2) = 7r 4 for circular functions. Notice also that these formulae, when established, imply that e1, e2, e3 are distinct, since if for instance e1 = e2 then we have (p(x + 1/2) - e 1 )(r(x) - e 1 ) = 0 which is impossible since r is not constant. Now we examine more carefully the reasons why e 1 , e 2 , e3 must be distinct. We know that {e1,e2,e3} C {a,/J,,}, and have deduced (3.40) from (3.39) which is correct if the sets are equal, and in particular if e1, e2, e3 are distinct. If any two of e1, e2, e3 are distinct, then again the sets are equal since the third elements are determined from e1 + e2 + e3 = a+ {J + 1 = 0. Hence the only possibility not accounted for is that all of e1, e2, e3 are equal (necessarily to zero) while {a,/J,,} is of the form {O,{J,-{J}, and so (3.36) becomes p' 2 = 4p(p 2 - {3 2 ). In this case we may still put y = 1/2, ,/2, (1 + ,)/2 in (3.37) when the three results corresponding to (3.40) reduce to the single equation

r(x + 1/2)p(x) = r(x + ,/2)p(x)

= p(x +

(1 + ,)/2)p(x) = -{3 2

BASIC ELLIPTIC FUNCTIONS

83

which is absurd since it implies for example that p(x + 1/2) = p(x + r/2) for all x, though ( T - l) /2 is not in the period lattice. To sum up, we have shown in all cases that the values e1 = p(l/2), e2 = p((l+r)/2), and e3 = p(r/2) are distinct, that the differential equation (3.36) may be written in the factored form (3.39), and that the results of adding half-periods to x is given by (3.40). In particular, on comparing the right sides of (3.36) and (3.39) we find

As an application of these results we can find an explicit formula for p(2x) as follows. We have from (3.37) if y---+ x that

p(2x)

= -2p(x) +

i (::(~?)

2 ,

2 and from (3.35) and (3.36) that p' = 4(p 3 + s 2 p - s 3 ) and r" = 6p 2 + 2s 2, where s2 = e2e3 + e3e1 + e1e2, s3 = e1e2e3 and of course e1 + e2 + e3 = 0. Hence writing p = p( x) we have 2

p(2x)

2

(3p + s2) = -2p+ ---- 4(p3 + s2p - s3)

(p 2 - s2) 2 + 8s3p ----, 4(p3 + s2p - s3)

(3.41)

which gives p(2x) as a rational function of p(x). The fact that each p(x) - ej, j = 1, 2, 3 is the square of a holomorphic function on C \ L will be fundamental to our treatment of Jacobian elliptic functions in section 5.1. It can be deduced as follows with some further manipulation. From (3.41) we have

(p(2x) - e1)r'(x)

2

(p 2 - s2) 2 + 8s3p- 4e1(p3 + s2p- s3) (p 2 - 2e1p ) 2 - (4ei + 2s2)P 2

+ 4(2s3 - e1s2)P + s~ + 4e1s3 2 - (2d + e2e3)] [(p(x) - e1) 2 - (e1 - e2)(e1 - e3)]2. [(p - e1) 2

(3.42)

Similar results obviously hold with e 1 , e 2 , e3 cyclically permuted. A second Weierstrass function ( is defined by integration of p( x) F2(x) - 82 to give

((x)

F1(x)

+ D2x

.!. +~~I ( x

DD m n

l

x+m+n,

_

l

m+n,

+

X

(m+n,) 2

(3 43)

)

·

·

84

ELLIPTIC FUNCTIONS

The general term may again be compared with those of LL I Jm + nrJ- 3 and so the double series is absolutely convergent. If we sum first over m, then over n, we have already shown that L I::(m + nr)- 1 = O and I: I::(m + nr)- 2 = 82 which shows that (3.43) is also correct (using the Eisenstein convention) if the summation is applied to the terms separately. Again results already established for F1 imply corresponding results for (. We shall be even more selective here, mentioning only the addition formula (3.30) which in terms of ( reads

((x + y)

1 ("(x) - ("(y)

= ((x) + ((y) + 2 ('(x) -

('(y) .

The function ( has the quasi-periodic properties

((x + 1) ((x+r)

Fi(x + 1) + 62(x + 1) = ((x) + 62, F 1 (x + r) + 62(x + r) = ((x) + r62 - 21ri,

and the values of ( at the half-periods are ((1/2) = 62/2, (( r /2) = -1ri + = -1ri+(l+r)6 2/2. The values of ( at quarter-periods i are found in exercise 7.10. when r The notation for r is not quite standardised. Abromowitz & Stegun in [AS], following [WW] write the periods of r as 2w1 and 2w 2 so that

r62/2, and (((l+r)/2)

=

Sometimes also the factor of 2 is omitted as in [Ah]. With this notation our r(z) r(zlr) r(z;l/2,r/2), or conversely r(z;w 1,w2) (2w1)- 1r(z/(2w1)lw2/w1). Also the numbering of e1,e2,e3 is not standardised. We have followed Hurwitz & Courant [HC] in putting e 1 r(l/2), e2 = r((l + r)/2), e 3 = r(r/2), which seems natural since it follows the anti-clockwise direction around the parallelogram with vertices 0, 1/2, (1 + r)/2, r/2; however other authors have e2 and e3 interchanged.

=

3.5

Exercises for Chapter 3

3.1 From (3.2) we have the limiting cases A(xlr)---+ Ek(x) as S'r---+ oo; in particular F1(xlr)---+ 1rcot(1rx) and F2(xlr)---+ 1r 2cosec 2(1rx). Similarly from (3.11) we have 62j(r)---+ 12 j as S'r---+ oo; in particular 62(r)---+ 1r 2/3. Hence for the Weierstrass functions we have ( compare exercise 1.5) r(xlr) ---+ £(x) := 1r 2cosec 2(1rx) - 1r 2/3, and ((xlr) ---+ Z(x) := 1rcot(1rx) + 1r 2x/3;

BASIC ELLIPTIC FUNCTIONS

85

these functions have not acquired generally accepted names, however. 3.2 (i) Show that e1 e2

= p(l/2)

= p((l + r)/2) 7r2

+

21r2

f(

1 _ 1 ) cos 2((n - 1/2)1rr) sin2(nH) '

1

00

3

+

21r2

I:

-= 2 - -1 - - - - - - - ,2 -1- - - ) .

(

sin ((n - 1/2)1rr)

1

sin (n7ri)

(ii) Deduce from (i) that if r = iy, y > 0 then e1 and e2 are decreasing, and is an increasing function of y. (iii) Prove corresponding results to (ii) when r = 1/2 + iy. In particular show that 8'e 2 -=/= 0 when r = 1/2 + iy and y E R is sufficiently large. 3.3 Show that e3

2 Fi(x - r/ )

.

~

1rsin(21rx)

= rn + ~ sin(1r(x + (n + 1/2)r)) sin(1r(x -

and find a similar series for F1(x - (1

(n

+ 1/2)r))'

+ r)/2).

3.4 Use (3.18) to show that 82(w) = 21r/V3, where w = (-1 + i../3)/2 = e21ri/3. 3.5 Show that for j ~ 2, r- 2io 2j(-l/r) = 82j(r). Deduce that D2j(i) = 0 when j is odd, and that D2j (w) = 0 unless j is a multiple of 3. 3.6 Show that F1(xlr) + Fi(x + 1/2 Ir) = Fi(2xl2r) and establish similar results for F1(xlr) + F1(x + (1 + r)/2 Ir) and F1(xlr) + Fi(x + r/2 Ir). 3.7 Substitute the power series (3.31) for p(x) into (3.35) to obtain the recurrence for 82 j(r), j ~ 4, (4j 2 - l)(j - 3)82j

I: (2/ -

=3

1)(2m - l)821D2m,

z+m=j Z,m;?:2

which determines the values of 82 j, j ~ 4 as polynomials in 84, 86. Use question 5 to deduce that when r = i (16j 2 - 1)(2j - 3)84j

=3

I: (4/ z+m=j Z,m;?:1

1)(4m - l)841D4m,

86

ELLIPTIC FUNCTIONS

and that when

T

=w

(36j 2

-

l)(j - l)c56j

=

I:

(61- 1)(6m - l)c56zc56m·

z+rn=j l,rn~l

(The values of c5 4 (i) and c5 6(w) will be found in section 7.5.)

4

Theta Functions In accordance with the plan of chapter 1, we introduce an infinite product

0. Then from (6.24) we have since 1 + c2 = 2(a 2 + b2 )/(a + b) 2 ,

G(a, b)

=

11r/

7r 2 --F(l/2, 1/2; 1; c2 ) = a+b a+b o 2 d0 2 --;:::======= 0 a + b) 2 - ( a - b) 2 sin 2 0

1rf 1 j(

2

d0

--;:::=== l - c2 sin 2 0

J

= G( a1, b1)

as required. As an example of the use of the AG M to calculate elliptic functions we could taker= i, k* = 1/-/2 when M(l, k*) is found to have the value 0.847213085 ..

146

ELLIPTIC FUNCTIONS

after 3 iterations. This is part of the striking chain of equalities

1

1/,/2) =

-;--1 Jf=t4 2v12

1

dt

f 2 (1/ 4) = 21rs/2 =

2

00

-n27r

(

~e

)

which follow on combining (6.26) with (6.10). Similarly if, = w + l = ei1r/ 3 , then k* = k = ei1r/ 6 and M(l, k*) is found to have the value 1r224/3 M(l ei1r/6) = ----ei1r/12 ' 31/ 4 f 3 (1/3) forming part of a similar chain of equalities; details are in section 7 .5. For general values of x, k 2 we can calculate sn(x, k 2) by first calculating , from the AGM and then finding sn(xJ,) as a quotient of theta functions.

6. 7

Exercises for Chapter 6

6.1 (i) Find the first few terms in the power series for the Jacobian functions about the origin. In particular show that u - 1! (1 + k 2 )u 3 + 1! ( 1 + 14k 2 + k 4) u 5 + ... , 5 3 1 2 3 1 - 16k 2 + k 4) u 5 + ... sdu u + ! (2k - l)u + 5!(1 3 (ii) Deduce from (i) that en is concave down on [O, K] if O < k 2 :=; 1/2, but has an inflexion in (0, K) if 1/2 < k2 < 1. 6.2 Deduce the rotation formulae en( iu, k 2) = nc( u, k* 2) and dn( iu, k 2) = de( u, k* 2) from (6.14) and (6.15).

snu

6.3 (i) Show that if k2 -+ O+ (corresponding to 8', -+ oo) then sn(x,k 2), cn(x,k 2), dn(x,k 2), K-+sinx, cosx, 1, 1r/2 respectively and find the limit lim

k 2 --,.0+

sn( x, k 2) - sin x k2

(ii) Show that if k 2 -+ L ( corresponding to 8',-+ O+) then sn(x, k2), cn(x, k 2), dn(x, k2),

I{-+

tanhx, sechx, sechx, oo.

6.4 Use (6.18) to find the expansion of the length of an ellipse in powers of q. (Combine the series for 0~ and e1 = 1r 2(0i+204)/3, with 82 = -41rir/(,)/17(,) and the expansion (4.14) for 17 = q1112 G.)

ELLIPTIC INTEGRALS

147

6.5 Lemniscate functions. The functions sl, cl (lemniscatic sine and cosine) were introduced by Gauss as the inverse of the integrals (1 - t 4 )- 1 12 dt, 1 2 4 t )- 1 dt respectively. Show that on the lemniscate (x 2 + y 2 ) 2 = a 2 (x 2 -y 2 ) we have (ds/dr) 2 = a 4 /(a 4 - r 4 ), and use the results of section 6.4 to deduce that

J;

J:(l -

(V'),sa '2!) '

en

_1 8 d

sl (~)

.,/2

(V2s !) a

'2

·

6.6 Define, corresponding to exercise 6.5, the functions se, ce (elliptic sine and cosine; discretion is advisable if evaluating the former at x) by

x

= a ce(s/a),

y

= bse(s/a)

where (x, y) is the point on x 2 /a 2 + y 2 /b 2 = 1 whose distance from (a, 0) is s. Find the first-order differential equations satisfied by se and ce. 6.7 Choose several values of rand make accurate versions of figures 13 and 14. The values T = 1/2 + i/(2./3), 1/2 + i/2 and 1/2 + i-/3/2 ( compare figure 23) are particularly interesting. 6.8 Show, corresponding to (6.16), that the addition formulae for en and dn are

cn(x + y) dn(x

+ y)

en x en y - sn x sn y dn x dn y 1 - k2sn 2 x sn 2 y dn x dn y - k 2 sn x sn yen x en y 1 - k2sn 2 x sn 2 y

(These may for instance be deduced from (6.16) by using cn 2 dn 2 = 1 - k 2 sn 2 .) k2

6.9 Suppose for simplicity that O + k*2 = 1. (i) Show that sn(K/2), cn(K/2), dn(K/2)


0 the mapping also preserves the real axis and the upper and lower half-planes. The natural correspondence between matrices and and bilinear maps is of particular importance because if we define Az = (az+b)/(ez+d) for a mat~ix

( ~ ! ),

then we easily verify that for any matrices A, B we have A(Bz) =

(AB)z; that is the composition of functions gives the same result as matrix multiplication. The correspondence is not one-to-one however since (

~

!)

-b ) both determine the same mapping. Hence we should and ( -a -e -d consider not SL(2, Z) but its quotient r := PSL(2, Z) = SL(2, Z)/{±1}. We shall however continue to regard the elements of r as matrices or bilinear mappings, rather then as equivalence classes; the identification of A and -A means that in examples we may assume that e > 0, or if c = 0 then d > 0. For the remainder of this section we shall investigate the structure of r. We begin by showing that r is generated by the mappings S, T or equivalently by the matrices (

~

-

~

) and (

~ ~

). Suppose then that we are given

a mapping f(z) = (az + b)/(cz + d) in r which we want to write as a product of powers of Sand T. If c = 0 then since a, b, e, dare integers with ad- be= 1 we must have a = d = 1 and so f = Tb. However if e > 0 then for any n we have r-n f(z) = (a - nc)z + (b - nd). ez+ d Hence if we choose n so that O ::; e1 = ne - a < e we shall have

sr-n f(z) =

CZ+

d

e1z + d1

which has a strictly smaller value O ::; e1 < c. Thus a finite number of repetitions of this process will give a mapping with e 0 which as we have

=

MODULAR FUNCTIONS

153

already seen corresponds to a power of T. Hence every element of f is a product of powers of S, T as predicted. Notice however that this representation is not unique without some further restriction since in addition to the obvious S 2 = I we have the less obvious ( ST) 3 = I; these relations determine the group completely. In addition to r we shall also be interested in the subset ~ of those matrices which, when we use (7.1) to map pairs of integers by ( :: )

( ~ ! ) ( : ),

preserve the parity of m, n. It is immediate that the

==

==

conditions a d 1 (mod 2) and b c 0 (mod 2) are necessary and sufficient for this to happen (exercise 7.2); equivalently a E ~ if and only if A= I (mod 2). The group~ is normal in r since it is the kernel of the group homomorphism which reduces the elements of A mod 2. In the terminology of general modular groups (for which see [Ko]), r is the full modular group and~ is its congruence subgroup of level 2. An argument (which we leave to the reader) analogous to the one given above for r shows that ~ is generated by the elements T 2 and ST 2 S(z) = z/(l-2z) or equivalently by the matrices

(~ i)

~

and ( -~ ) . Since ad - be= l, an arbitrary element of r must have one of the forms

where e and o denote even and odd integers, and so must be congruent mod 2 to exactly one of the matrices

(7.2) These matrices are of course not a subgroup of r but determine the coset decomposition of ~ in r, so that ~ is a subgroup of index 6 in r.

7.2

Fundamental Regions

We found in section 7.1 that the group r acts on the upper half-plane H in such a way that (AB)z = A(Bz) for all A,B Er, z EH. We shall say that points z, z' E H are congruent mod r if there is some A E r such that z' = Az. We write z z' (mod f) when z, z' are congruent mod r. Congruence is an equivalence relation because of the group structure of r, and it is important

=

154

ELLIPTIC FUNCTIONS

to determine the equivalence classes. A fundamental region is a connected set which contains one element from each equivalence class. These definitions apply with obvious modifications if r is replaced by one of its subgroups such as~We begin with r and go on to show how results for ~ may be deduced. Since for z E H, n E Z we have z z + n (mod r) we may restrict attention to a strip of width 1; the choice R := {z: -1/2::; ~z < 1/2} is customary.

=

-1

0

=Argz, a =min (8, Jt -

Figure 16

0)

Mapping by S

Our aim is to show that the part of R with lzl 2: 1 (suitably modified on the boundary) is a fundamental region for r. If z ER, lzl < 1 then Sz -l/z has strictly greater imaginary part, and the same is true for rn Sz where n is chosen to bring S z back into R. Moreover if we put a := min( arg z, 1r - arg z) and O < a ::; 1r / 6 then referring to figure 16, ~ S z is greater than ~ z by a -1/2 is mapped by S onto the circle factor of at least 3, since the line ~z lz - 11 = 1. Hence a finite number of applications of S, T will bring us into that part of R with 1r /6 ::; a ::; 1r /2. But if 1r /6 ::; a ::; 7r /3 then S z lies in the region above all circles lz - nl = 1, n E Z, since the line through the origin and w + 2 is tangent to the circle lz - 21 1 at w + 2, and the same is evidently true if 7r /3 ::; a ::; 7r /2. Hence after one further step we reach the subset of R

=

=

=

MODULAR FUNCTIONS in which

lzl

~ 1.

155

At this point it is convenient to introduce two sets

{z: -1/2::; ~z::; 0, lzl ~ 1}, { z : 0 < ~z < 1/ 2, Iz I > 1}

G1 G2

which are closed and open respectively in H, as indicated in figure 16. We shall show that G := G1 U G2 is the required fundamental region. So far we have shown that every point of H is congruent to some point of G. But the sides of G where ~z ±1/2 are obviously equivalent by z _,. z + l, and the parts of the boundary where lz I = 1 and 1r/3 ::; arg z ::; 1r /2, 1r /2 ::; arg z ::; 21r /3 are equivalent by z _,. -1/ z. Hence every point of H is equivalent to a point of G. It remains to show that distinct points of G are inequivalent; equivalently that if z E G, A E r, then either Az (/:. G or Az = z. For this we observe that we can write Az = (az + b)/(cz + d) in the form z + b if c 0 or [a - l/(cz + d)]/c if c i- 0. In the first case, if z E G and b i- 0 then z + b (/:. G. In the second case if z E G we have icz + di ~ 1 and so a - 1/(cz + d) lies on or below the circular arcs lz - nl = 1, and then the same is true for [a - 1/( cz + d)]/ c. If z is in the interior of G then Az will be strictly below the arcs iz - nl 1 and so not in G, and the same is true for z on the part of the boundary with ~z = -1 /2, Iz I > 1. If z is on the part of the boundary with lzl = 1, and if c ~ 2, then 1/(cz+d) is again strictly below the arcs jz - nl = 1 and so not in G; the same is true if c = 1, d i- 0 when for lzl = 1, z i- i, w we have Az = a - 1/(z + d) (/:. G. If z = w then Aw = w for a = c d = l, but if z = i then Ai i- i for any d i- 0. The only case remaining is when c = 1, d = 0, when Az = a -1/z can only be in G if either z = w, a= -1 when Az = z = w, or when z = i, a= 0 when Az = z = i; in each case we have found fixed points of mappings of r. Hence for any z E G, if Az i- z then Az (/:. G as required. When we consider fundamental regions for subgroups r 1 of r it is easy to show that if we enumerate the cosets of f1 in r as {a 1f 1, a2f1, ... , arfi} where r 1 has index r in r then a 1 GU a 2 G U ... U arG is a fundamental region for f 1 · Applied directly in the case r 1 = ~ with a1, ... a6 as the matrices of (7.2), this gives a slightly untidy result as the reader may verify, and it is more satisfactory to separate G into G 1 and G2 and consider the twelve regions given by {G 1, T(G 1), S(G 1), r- 1s(G 1), ST(G 1 ), TST(G 1 )} which are closed, and {G 2, r- 1(G 2), S(G 2 ),TS(G 2 ), sr- 1(G2), r- 1sr- 1(G1)} which are open, as indicated in figure 17. We can tidy up the boundary by moving the line segment [1 + i, 1 + ioo) to [-1 + i, -1 + ioo) by r- 2 z = z - 2 E ~ and moving the circular arc from 1 to (1 + i)/2 to the arc from -1 to (-1 + i)/2 by ST 2 Sz = z/(1- 2z) E ~This gives us D := D1 U D2 where

=

=

=

=

{z: -1::; ~z::; 0, lz+ 1/21 ~ 1/2, S'z > O},

156

ELLIPTIC FUNCTIONS

D1 and D2 partitioned

Figure 17

D2

{z: 0 < ~z < l, lz - 1/21 > 1/2, S'z > O},

and since the elements of each of the sets of six mappings considered above come one from each coset of ~ in r it follows that D is a fundamental region for~-

7 .3

The Elliptic Modular Function

The function k 2 ( r) = 0j'(Olr)/0i(Olr) was defined in section 6.2 as the modulus of the Jacobian elliptic functions; it is usually referred to as the elliptic modular function. Since k 2 is a cumbersome name for a function we shall replace it with the traditional ..\,

..\(r)

:=

k 2 (r) for S'r > 0.

We begin by finding the effect on A of the transformations T : r ~ r + l and S : r ~ -l/r which generate the modular group. These results follow from our work in chapter 4. We know that if r ~ r + l then q = ein ~ -q, and from (4.13) that 0i(O I r) = l6q TI~(l - q2 n)4(l + q2 n) 8 ~ -0i(O I r)

MODULAR FUNCTIONS and 0j(O we have

Ir)= Il~(l -

A(Tz) = A(r+ 1)

157

q 2 n) 4 (1

= k2(r+ 1)

+ q2n- 1 ) 8 =

-,.

-04(0

Ir).

Hence from (4.19)

0i(O I r + 1) -0i(O I r) 0j(O I T + 1) - 0j(O I r) - 0i(O I r) A(r) (7.3) A(r)-1·

We can apply this repeatedly to obtain A( r + n) A( r) if n is even, A(r + n) = A(r)/(A(r) - 1) if n is odd. In the same way if r-,. r* = -1/r then from (4.27), 0i(O Ir)-+ - r 2 04(0 Ir) and 01(0 Ir)-+ - r 2 04(0 Ir), so

A(Sz)

0t(o I r)

= A(-1/r) = 0j(O Ir) = 1 -

A( r).

(7.4)

From (7.3) and (7.4) we deduce that A(T2 z) = A(ST2 Sz) = A(z) so that A is invariant under ~. The six mappings determined by {z, 1- z, 1/ z, 1/(1- z), z/(z -1), 1-1/ z} form the anharmonic group which we shall denote by E. It is important in projective geometry where it gives the possible values of a cross-ratio. It is generated by the two mappings z -,. z/(z - 1) and z -+ 1 - z but is not a subgroup of r since not all its elements are in r (see also exercise 7.3). We deduce from (7.3) and (7.4) that for any A Er there is f E E such that

A(Az)

= f(A(z)).

(7.5)

To investigate the mapping properties of A we start from the fact which was established in section 6.1 that the quotient 02 /0 3 decreases monotonically from 1 to O when r moves from O along the imaginary axis in the direction of increasing S'r. Thus .:\ is a bijection from ( 0, ioo) to ( 0, 1) and in particular A(i) = 1/2. From this and (7.3) it follows that.:\ is a bijection on either of the intervals (±1, ±l+ioo) to (-oo, 0) with .:\(±l+i) = -1. But r-,. -1/r maps (-1, -1 + ioo) on to the arc of the circle lz -1/21 = 1/2 with S'z > 0, so from (7.4) we see that.:\ maps this arc bijectively to (1, oo) with .:\((1 + i)/2) = 2. Since in D 2 , .:\( r) -+ 0, 1, oo as r -+ ioo, 0, 1 respectively, we have shown that .:\ maps the boundary of D2 onto the real axis, as shown in figure 18. As 2 8 2 8 ~T -,. oo in D2, q = ei1rr -+ 0 so .:\( r) = 16q TI~ (1 + q n ) (1 + q n-l )- -+ 0 uniformly as S'r-+ oo. Also as ~T-+ oo, -1/r-+ 0 in D 1 and 1-1/r-,. 1 in D2. It follows that if r-,. 0 in D1 (or D 2 by reflection) then .:\(r)-+ 1 from (7.4) and if r-+ 1 similarly then .:\(r) -,. oo from (7.3). The rate at which .:\( r) -+ 1 as T -+ 0 is exponential since

158

ELLIPTIC FUNCTIONS

A(l+i)=-1

A(oo)=O

Figure 18

A(z)= 1/ 2

A(O)=l

The mapping by ). of D 2 onto H.

A((l+z)/2)=2

MODULAR FUNCTIONS

159

as 1 ----+ 0 in D2. In fact this behaviour is repeated at all rational boundary points of H, with >.(1) tending rapidly to one of 0, 1, oo (exercise 7.4). This behaviour demonstrates the existence of the natural boundary of the function on the real axis since there can be no function which is analytic on any open set containing a subinterval of R which agrees with ,\ on the intersection of their domains. Other modular functions have similar natural boundaries on the real axis; we shall investigate the behaviour of 03 in detail in section 8.3. Our objective is to show that ,\ extends to a holomorphic bijection of D 2 onto H. We shall give a proof which builds on properties of elliptic functions established in earlier chapters; another proof using the argument principle may be found in [Ah]. The most important step in the proof is to show that ,\ is one-to-one on D2 for which we use the following argument, essentially due to Dedekind. Suppose that we have ki = >.( 11) = >.( 12) = ki for points 11, r2 E D2. Then the Jacobian functions fi(x) = sn(x, ki) and f 2 (x) = sn(x, ki) satisfy the same differential equation (!') 2 = (1 - f2)(1 - k 2 f 2 ) and so (!{) 2 = (!~) 2 wherever both exist. But f 1 , h are analytic and non-zero in C except for zeros at points of L1 = L(I{1 , !{;), L 2 = L(I{2, I 0, - { ei(n-l)1r/4(~) e-im1r/4 (~)

a(m, n) -

if n is odd, if n is even.

(8.24)

We already know that for m, n > 0 (the conditions (m, n) = 1, 2lmn will be assumed throughout without further mention) we have the reciprocity and conjugacy relations ) a ( m,n ) _- e -i1r/4 a ( -m,n ) -_ e -i1r/4 a ( m,n,

and the periodicity relations a(m + 2kn, n) = a(m, n) = a(m, n

+ 2km).

(8.25)

For initial values we have a(m, 1) = G(m, 1) = 1 for even m > 0, and so a(l, m) = e-i1r/ 4 _ Also a(m, 2) = G(m, 2)/,/2 = e':fi'Tr/ 4 as m = ±1 (mod 4), and so a( 2 ,m)={ 1_ ~fm=l (mod4) -z 1f m = -1 (mod 4). We shall deduce the general value (8.24) from these initial values by induction; note that the values that we have already found are in agreement with the result. For the inductive step it is convenient to extend the result to include the cases when either m or n (but not both) are negative. The result to be proved then reads ei(n-l)1r/ 4 (

a(m,n) =

{

r;: ~

1 1

n e -im1r/4 ( TmT

if n is odd, · even, 1·f n 1s

(8.26)

and it is a routine exercise to check that the right side of (8.26) satisfies the same reciprocity, conjugacy and periodicity properties as does a. (The periodicity condition (8.25) is then valid for m, n > 0 and any k E Z.)

APPLICATIONS

187

The proof of (8.26) now proceeds by induction on the smaller of m, n; we have already established the cases when this is 1 or 2, and this extends at once to the case when one or other (but not both) are -1 or -2. Suppose that we have established the result for min( m, n) = 1, 2, · · ·, N say. Then for any m, n with min(lml, lnl) = N + 1, say m = N + 1, reduction of n modulo 2m brings us to a value n' say in the range -N :=:; n' :::; N for which the result is known. (We cannot come to the point (N + 1, ±(N + 1)) since N + 1, ±(N + 1) are not coprime.) Hence the result is proved for min( m, n) :=:; N + 1 and so generally by induction.

07 02

07

0 l 8

02

06

7

00 03

06

00

0

00

05

06 05

04 03

00 0 1

02 01

07

0 l 2

02

04

06

3

0 1

03

07

0 l 4

07

06

00

5

02

07

07

0 l 6

04

01

00

06 0 1

07 00

00

00

7 -4

-3

-2

-1

2

0 6/2

0612

0

7/3

0 0 4/0

00/4 0

3/7

3

06 5/1

-2

07

0

5/1 -4

07

Figure 20

5

6

7

06 07

o5 00

-3

4

06

o5

o4 01

8

06

oO 03

05

Values of a.

Figure 20 shows the values of a for -4 :=:; m, n :=:; 8. The value plotted is s where a(m, n) = eiS1f/4, 0 :=:; s :=:; 7. Note the conflict of values if we use periodicity to extend the definition to the third quadrant where both m, n < O; a displays a kind of behaviour for integer values of m, n which corresponds

188

ELLIPTIC FUNCTIONS

vz

to the branch point of at z = 0, where to return to the initial values it is necessary to continue twice around the origin .

8.4

Elliptic Curves

The study of elliptic curves arises from one of the oldest parts of number theory, namely the finding of solutions in integers for algebraic equations such as y 2 = x 3 - 2, and has developed in the last few decades into one of the most active areas of mathematical research, involving sophisticated methods and concepts from many branches of the subject. Our aim in this section will be to give an introductory account of such parts of the theory as are accessible using the results we have developed in chapters 3-7, and to indicate where the deeper theory is to be found. The results depend greatly on the field in which the points of the curve are supposed to lie and it is natural for us to begin with C and then specialise first to R and then to Q. However any field F contains a unique homomorphic image of Z determined by lz ......,. lF, and so when the coefficients are integers it is helpful to regard the curve as something which is independent of any particular field and which appears in various forms when different fields are specified. 8.4. 1

Elliptic Curves over C

For our purposes, a complex elliptic curve P is a curve in C 2 which is parametrised by a Weierstrass elliptic function. More explicitly for z E C, define (x, y) = p(z) := (r(z), r'(z)); the differential equation (3.36) gives the canonical equation of the curve in the form (8.27) Notice that when z ......,. 0 both x, y ......,. oo and so we must include a point at infinity on the curve; this can be thought of either as the unique point at infinity on the Riemann sphere, or more usefully as the point (0, 1, 0) when we consider the curve in homogeneous projective coordinates ( u, v, w) in the form v 2w = 4u 3 - g2uw 2 - g3w 3 . [This drastic way of defining an elliptic curve sidesteps the whole business of algebraic curves and their genus. There is some compensation however in that it justifies calling the curve elliptic when as we shall see the geometrical form of the curve is anything but elliptical. For those in the know, an algebraic curve of genus 1 over a field F whose characteristic is not 2 or 3 can, if it contains a single point whose coordinates are in F, be reduced by a birational transformation, to the Weierstrass canonical form (8.27) with g2, g3 E F. The relevant definitions and details of the reduction are in [Coh] or [Fu]; two special cases of the reduction are in exercises 8.9 and 8.10.]

APPLICATIONS

189

A general algebraic curve of the form (8.27) is said to be non-singular if the cubic polynomial 4x 3 - g2 x - g3 has distinct roots, which we have shown in exercise 7 .7 occurs if and only if .6. = g~ - 27 gJ f O; this is of course satisfied when Pis parametrised by (r, r'). We showed in section 7.4 that for any 9 2 , g3 with .6. f O there exists a lattice L(w1,w2) such that 92,93 = b4,b6(w1,w2) and hence every polynomial equation of the form (8.27) with .6. f O can be parametrised by (x, y) = (r(z), r'(z)) for a Weierstrass function over some lattice L. The lattice L of periods of r is a subgroup of C and so the mapping p : z - t (r(z), r'(z)) induces a bijection of C/ L onto the curve P. Since C / L is isomorphic as an additive group to T 2 ( the two-dimensional torus, where T = R/Z), the curve inherits a group structure from T 2 defined by

P1 + P2

:=

(r(z1 + z2), r'(z1 + z2))

where Pi = (r(zj ), r'(zj )), j = 1, 2. [It is also worth noting that the Riemann surfaces of both z 2 = 4w 3 g2 w - g3 and z 2 = (1- w 2 )(1- k 2w 2), k 2 f 1 are topologically tori. Then the elliptic integrals of chapter 6 are the integrals of the unique holomorphic form dw/ J(l - w 2 )(1 - k2w 2 ) over the generators of the torus; more concretely they are the values of f dw / y taken over the relevant portion of the elliptic curve y 2 = (1 - x 2 )(1 - k 2 x 2 ) which is reduced to canonical form in exercise 8.10. There is an interesting discussion relating this to the Landen transform (exercise 6.l~nd the arithmetic-geometric mean in [AMM]. The Riemann surface of y1p(z) for a polynomial pis described on pages 157-163 of [JS].] The neutral element of the group P is the image of zero in C, namely the point at infinity on P (this form of words avoids the embarrassment of having to say that on P, zero is at infinity!); also the inverse of p = (x, y) is -p := (r(-z), r'(-z)) = (x, -y). The group structure on P gains greatly in importance and interest when we observe that it can also be interpreted geometrically:

P1 + P2 + p3

= 0 if and only if the points P1, P2, p3 are collinear in C 2.

(8.28) This is in fact simply a restatement in the present context of the addition formula for r, since the condition for Pi = p( Zj), j = 1, 2, 3 to be collinear is that

r'(z3) - r'(z1) _ r'(z2) - r'(z1) r(z3) - r(z1) - r(z2) - r(z1) '

or equivalently from (3.37) that

which is satisfied when z1 + z2 + Z3 E Lor equivalently Pl+ P2 + p3

= 0 on P.

190

ELLIPTIC FUNCTIONS

Condition (8.28) can be restated more constructively by saying that if

=

Pl, P2 E P and the chord joining P1 to P2 (the tangent at p 1 if p 1 p2 ) meets the curve again at q then Pl + P2 -q, the reflection of q in the x-axis. The group has the algebraic property of divisibility, that is for p E P and

=

> 1 there is q E P with nq = p. To see that this is so, let (p(z), r'(z)) when there are n 2 possible values of q, given by + kr)/n) for O ~ j, k ~ n - 1. Values at some of these division

any integer n

= p(z) = q = p((z + j

p

points were found in exercise 7.10. For computational purposes it is convenient to rescale, replacing ( x, y) by (x/4, y/4) so that (8.27) becomes y 2 = x 3 +ax+ b where a= -4g2, b = -l6g3.

(8.29)

In this notation, given points P1 = (x1, Y1), P2 = (x2, Y2) on P, the algorithm for finding p3 = (x3, y3) =Pl+ P2 is as follows. The slope of the line P1P2 is given by m := (Y2 - Y1)/(x2 - x1) when x1 # x2, otherwise if x1 = x2 then either P1 = -p2 when P1 +P2 = 0, or P1 = P2 when the tangent at P1 has slope m := (3xr+a)/(2y1) from (8.29). The third intersection of y-y 1 = m(x-x 1) with P is when (Y1 + m(x - x1)) 2 = x 3 +ax+ b and since also

Yi = xf + ax1 + b we have

But m is chosen so that one solution is x 2 so that also

and hence m 2 = x + x1 + x2 by subtraction. Thus X3 = -x1 - x2 + m 2 and -y3 - Yl = m(x3 - x 1) (note the minus sign for y3). In summary, for Pl f:. -p2, we have the following formula for (x3, y3), the sum on P of (x1, Y1) and (x2, Y2). m

{ (Y2 - Y1)/(x2 - x1) (3xr + a)/(2y1)

X3

m2 - X1 - X2,

Y3

-Yi - m(x3 - x1).

if X1 if X1

f:.

X2,

= X2, (8.30)

To illustrate, consider the curve y2 = x 3 - 4 which has the obvious solution (2, 2). If we take Pl = P2 = (2, 2) then m = 3, p3 = (x3, y3) = 2p1 = (5, -11). For 3p1 = 2p1 + Pl we find m = -13/3, x = 106/9 = and y = 1090/27 = 40 ~~. For 4p1 = 2p1 + 2p1 we find similarly m = -75/22, x = 785/(22) 2 = 1.6219 ... and y = -5497 /(22) 3 = -0.5162 ....

lli

APPLICATIONS

191

This example illustrates two features of calculation on elliptic curves. Firstly, given a point in C 2 , its multiples will lie in different translates of the fundamental parallelogram in C 2 and so the corresponding points of the curve will tend to jump around unpredictably. Secondly when dealing with rational points there is a tendency for the denominators to grow rapidly so that it may be better to rewrite the algorithm in homogeneous form so that all calculations are in integers. It is possible to define an elliptic curve simply by the algebraic equation (8.29) with 4a 3 + 27b 2 # 0 and the group operation given by (8.30). However this makes verifying the group axioms (particularly the associative law) a tedious chore. The advantage of defining the curve as the image of C / L by the map z -+ (r( z), r' ( z)) is that the group properties are inherited directly from those of C / L. 8.4.2

Elliptic Curves over R

According to our preliminary discussion, we could consider a real elliptic curve to be the points of a general ( complex) elliptic curve whose coordinates are in R. However this is not quite what we want, since the intersection with R 2 may be a finite set; for instance y 2 = x 3 + i has no real solutions, and y2 x 3 +ix+ l has only (0,±1). To obtain a genuine curve in R 2 we require infinitely many real points; evidently if there are at least two real points with distinct x-coordinates on y 2 = 4x 3 - g2x - g3 then both g2, g3 are real and conversely if g 2, g3 are real then for sufficiently large x, 4x 3 - g2x - g3 is positive and for these x there are real values of y. Hence we define a real elliptic curve by an equation of the form y 2 = 4x 3 - g2x - g3 with g2,g3 E R and ~ = g~ - 27gj # 0. Since we have parametrised our elliptic curves by a Weierstrass function r we should investigate what conditions on r or its underlying lattice result in real values of g2, g3. When g 2 , g3 E R we can deduce from the recurrence relations (exercise _11l that all coefficients in the power series (3.31) for rare real so that r(z) = r(z) and in particular r(t), r'(t) are real for real t. But if r(z) = r(z) then since r has a pole at each point of the lattice we must have L = L. In this case, if z E L then z E L and z + z E L so L must have a pair of generators w1, w2 7W1 with O < w1 E Rand 8'7 > 0. But then 7w 1 must be in the lattice and so there are integers m, n with rw 1 = mw 1 + n7w 1. Equating real and imaginary parts gives n = -1 and 2~7 = m so 2~7 E Z is necessary for the function r( z I7) to define a real curve, and it is easy to reverse the above argument to show that it is also sufficient. Hence all real elliptic curves fall into two classes according to whether ~7 E Z (typically ~7 = 0) or ~7-1/2 E Z (typically ~7 = ±1/2). When ~7 0 the fundamental parallelogram with vertices 0, 1, 7 + 1, 7 is a rectangle, typified by the lemniscatic case, 7 i when the parallelogram is a

=

=

=

=

192

ELLIPTIC FUNCTIONS

square. When ~' = 1/2 the fundamental parallelogram with vertices 0, r, 1,, is a rhombus, typified by the equianharmonic case, , = ei 11-f 3 = w + 1 when the parallelogram is formed by two equilateral triangles. We consider these cases separately. In the rectangular case, q = ein is real and positive and from the product formulae (4.13) 0j(O), j = 2,3,4 are real and positive with 02 ,03 increasing with q, and 04 decreasing. Hence from (7.8) we deduce that e1 > 0 > e3 and e1 > e2 > e3 with e2 being positive, zero or negative as g3 is < 0, = 0, > 0, or correspondingly 8-, < 1, = 1, > 1; these intervals are labelled a and b in figure 23. Evidently ~ > 0 since all roots are real, and the curve consists of two disconnected pieces, one bounded where e3 :::; x:::; e2 and one unbounded where x 2: e1. The unbounded part is the image of (0, 1) by t-+ (r(t), r'(t)), and the bounded part is the image of (, /2,, /2 + 1) where r, r' are real using periodicity and r(z) = r(z). Figure 21 gives the shape of these curves. When 8-,-+ oo, q-+ 0 we obtain the limiting singular case in which 02 -+ 0, 03, 04-+ 1 so e1-+ 271" 2/3, e2,e3-+ -71" 2/3 and the curve is y 2 = (x-2)(x+1) 2 (after rescaling). Similarly when 8-,-+ O+, q-+ L we find 04 -+ 0, 02 -+ oo and 02 - 03 -+ 0 so the curve is y 2 = (x + 2)(x - 1) 2 (also after rescaling). And if in either of these cases we allow all of e 1, e2, e3 -+ 0 we obtain the degenerate case y 2 = x 3 . These curves are shown in figure 22. The three types of singular curve are referred to as being of split multiplicative, non-split multiplicative, or additive degeneracy, according to whether they can be written in the form y 2 = (x - a)2(x - b) (where always 2a + b = 0) with a > b, a < b or a = b = 0 respectively. When we encounter this again in the context of curves over finite fields such as F P the condition a > b or a < bis interpreted as saying that a-bis or is not a square in the field. The reason for the exotic-sounding names lies in the topological structure of the curve minus the singular point; we investigate this in the exercises. In the rhombic case we found in section 7 .5 that if , = ei-rr / 3 then g2 = O,g3 > 0 and so~= g~ - 27gi < 0. But g2,g3 are real so~ must be real, and we know that ~ is never zero in H so that ~ must be negative on the whole of ~' = 1/2. Hence the roots of the cubic polynomial can not all be real and so must consist of one real root and a conjugate pair. However r(t) is real for real t so e1 = r(l/2) is real and e2 ,e3 are the conjugate pair. It follows that the curve consists of a single connected component, the image of (0, 1) by t-+ (r(t), r'(t)). From g3 = 4e 1e 2e 3 we see that the sign of g3 is the sign of e1, and from g~ = J ~ that the sign of g2 is opposite to the sign of J. We show in exercise 8.11 that e1 is an increasing function of y when , = 1/2 + iy, y > 0. But g3 = 0 when , = i (the lemniscatic case) and so also when , = (1 + i)/2 which is conjugate to i by r and it follows that e1 and g3 are positive for 1/2 < y < oo (labelled d and e in figure 23) and negative for O < y < 1/2 (labelled f and g). For J we already know the values on the boundary of G, namely that O < J < 1 on the unit circle, and O > J > -oo

APPLICATIONS

193

Graph of y = 4x3- g 2 x- g 3

(ii) g 2 >0, g 3 =0, lm,:;=l, Lemniscatic case

Figure 21

Rectangular elliptic curves.

ELLIPTIC FUNCTIONS

194

Graph of y = 4x3- g 2 x- g 3

(i) g 2 =12e 1 2 , g 3 = -8e 13 , e 1 >0. Split multiplicative case

(ii) g 2 =g3 =0. Additive case

(iii) g 2 =3e 1 2 , g 3 = e 13 . Non-split multiplicative case

APPLICATIONS

195

on ~T = 1/2. Transferring this information to the line ~T = 1/2 (figure 23 again) we deduce that J(l/2 + iy) is negative for y > ,/3/2 (labelled d) and 0 < y < 1/2,/3 (labelled g), and is positive for 1/2,/3 < y < ,/3/2 (labelled e and !) . This determines the signs of g2, g3 at all points of ~T = 1/2. The variety of forms that the corresponding curve may take is larger than in the rectangular case since it depends on the signs of both g2 and g3 • Referring to figure 23, which shows the regions into which H is partitioned by r, we see that there are seven possibilities giving rise to the elliptic curves in figure 24 as indicated. The property of divisibility now depends on whether the curve is rectangular or rhombic; evidently in the rhombic case when the curve is simply the image of (0,1) then for any point p there will be n possible values of q with nq = p, just as there were n 2 values in the complex case. In the rectangular case, points on the bounded component which is the image of ( r /2, r /2 + 1) will not be divisible by 2, though points on the unbounded component will be divisible by any positive integer. Points of order 2 (i.e. those with 2p = 0) are where the tangent is parallel to the y-axis and these are evidently the points ( ej, 0) whenever these are real. Points of order 3 (3p = 0) are given by the real points of inflexion since the tangent at such a point intersects the curve three times there. We show in exercise 8.12 that a real elliptic curve always has exactly two points of inflexion, both of them on the unbounded component. 8.4.3

Elliptic Curves over Q

It is when we consider elliptic curves over discrete fields such as Q, or finite fields such as Fp Z/pZ, which are of most interest for applications in arithmetic, that the problems of greatest difficulty arise. Many of the results are deep and we shall be content to give references to the literature. In particular the reader may consult the monograph of Cassels which develops many of the most interesting results without assuming a wide background. We begin with the obvious assertion that, just as for real elliptic curves, if the curve y 2 4x 3 - g 2x - g3 contains two rational points with distinct xcoordinates then g2 , g3 E Q. The condition that g2 , g3 E Q does not however imply the existence of rational points on the curve since for instance it can be shown that, other than the point at infinity, the only rational solutions of 6y 2 = x 3 - 2 are (2, ±1) (see exercise 8.9 on Fermat's equation), and that y 2 x 3 -9 has no rational solutions at all. (Here and throughout this section, x 3 + k, k E Z are taken assertions about the rational points on curves y 2 from the tables in [Ro] which in turn come from (Ca2].) Given a curve P : y 2 4x 3 - g2x - g3 with g2, g3 E Q we can obviously

=

=

=

=

=

ELLIPTIC FUNCTIONS

196

Figure 23

Subsets of

~T

= 0,

~T

= 1/2.

APPLICATIONS

197

clear the denominators and rescale to obtain the curve in the form y

2

= x 3 +ax+ b,

a, b E Z,

~ = 4a 3

+ 27b 2

(8.31)

which we shall take as our standard form for an elliptic curve from now on. Since the group law given by (8.30) is rational in the coordinates x, y, the rational points form a subgroup of P which we shall denote by I{. When looking for rational points it is interesting to look first for points of finite order. We have already found that the points where the tangent passes through the point at infinity are of order 2, and such a point is rational if and only if the corresponding root of x 3 + ax + b is rational. Similarly the two real inflexions will be rational points of order 3 when their coordinates are rational. Points of order 4 and 6 can be found by looking for rational points whose tangents pass through the points of order 2 or 3 already found. For instance the reader can check that the tangent at (4, 9) on the rhombic curve y 2 = x 3 + 6x - 7 (x - l)(x 2 + x + 7) passes through the vertex at (1, 0) and so the point (4, 9) has order 4. On the rectangular curve y 2 = x 3 - llx + 14 = (x - 2)(x 2 + 2x- 7) the tangents from both (1, ±2) and (3, ±2-/2) pass through the vertex at (2, 0) so these are all real points of order 4 though only the first pair is in Q. For an example of a point of order 6, take any integer r, and consider the point p = (2r 2 , 3r 3 ) on the curve y 2 = x 3 + r 6 . This has 2p = (0, - r 3 ) which is a point of inflexion of the curve. In fact the number of possible rational points of finite order, and their degrees, is severely limited as we shall see. If p, q are points of order m, n then p + q has order at most mn and so the points of finite order form a group, the torsion subgroup T of the group I{ of all rational points on C. A useful theorem of Nagell and Lutz says that points (x, y) of finite order on (8.31) must have integer coordinates and that y 2 l4a 3 + 27b 2 . Hence to find T we can begin by enumerating the integers y with y 2 l4a 3 + 27b 2 and find for each whether there is x E Z with y 2 = x 3 +ax+ b; an integer solution of this must be a divisor of b - y 2 which gives an effective bound for the calculation. This is not the end however since the example considered in section 8.4.1, in which p = (2, 2) on y 2 x 3 - 4 and 3p is nonintegral, shows that an integer point may have infinite order. Thus having found a list of integer points with y 2 l4a 3 + 27b 2 we should check each of them to see whether it has finite order. For this we can use the fact that the order of a point, if finite, can only have one of the values 2,3,4,5,6,7,8,9,10 or 12. This is a consequence of a difficult theorem of Mazur, which says that the torsion group T must have one of the forms Z/nZ with 1 :::; n :::; 10 or n = 12, or Z/2Z x Z/2nZ with 1 :::; n :::; 4, and that each of these occurs infinitely often. Thus to find whether a given integer point has finite order the following simple proceedure is sufficient.

=

=

198

ELLIPTIC FUNCTIONS

Graph of y = 4x3- g 2 x- g 3

(i) g 2 >0, g 3 >0, lmT>"'13/2. Note e 1 is a decreasing function oflmT

(ii) g 2 =0, g 3 >0, lmT="'13/2. Equianharmonic case

APPLICATIONS

Graph of y

= 4x3-

199

g 2 x- g 3

(iv) g 2 0, (iii)

a< 0. In case (i) there is a triple point at (0, 0). Show that points with parameters

t 1 ,t 2,t 3 are collinear if and only if 1

1

1

-+-+= 0, t1 t2 t3 and deduce that the mapping t ---+ 1/t gives an isomorphism of the curve with (R, +), the real numbers under addition. In case (ii) let k := v'3a, the positive slope of the tangent at (a, 0). Show that points with parameters ti, t2, t 3 are collinear if and only if

(t1 + k)(t2 + k)(t3 + k) _ l (t1 - k)(t2 - k)(t3 - k) - ' and deduce that the mapping t ---+ ( t + k) / (t - k) gives an isomorphism of the curve with (R \ {O}, x ), the non-zero real numbers under multiplication. In case (iii) let k := ,;=Ja. Show that points with parameters t 1 ,t 2 ,t3 are collinear if and only if

(t1 + ik)(t2 + ik)(t3 + ik) _ l (t1 - ik)(t2 - ik)(t3 - ik) - ' and deduce that the mapping t---+ (t + ik)/(t - ik) gives an isomorphism of the curve with (T, x ), the unit circle under multiplication.

210

ELLIPTIC FUNCTIONS

The form of the curves explains why we referred to these as the additive, split and non-split multiplicative cases in section 8.4.2. Note that in case (i) there are no points of finite order, in case (ii) there is only one point of finite order at (-2a, 0) where t = 0, while in case (iii) there are n distinct solutions to np = 0 for all n ~ l. 8.14 We showed in section 8.4.3 that if all rational points on an elliptic curve are integral then the rank must be zero, and applied this to the curve y 2 = x 3 - x. Apply the result to some other curves of the form y 2 = x 3 - n 2 x. 8.15 (i) The equation a 4 + b4 = c2 has no solutions in integers with abc -:fa 0. (The proof is by descent-that is we assume a counter-example exists and use it to construct another with a smaller value of c.) Fill in the details of the following sketch. First since ( a2, b2 , c) is a Pythagorean triple, there are r, s with r2

s

-

2

= a2 ,

2rs

= b2 ,

r2

+ s 2 = c.

Then ( a, s, r) is a Pythagorean triple so there are t, u with t

2

-

u

2

= a,

2tu

= s,

t

2

+ u 2 = r.

It follows that b2 = 4tu(t 2 + u 2 ) and since (t, u) be squares, giving

t

= ti,

U

= ur,

t 2 + u2

= 1, all oft, u, t 2 + u 2

must

= tf + uf = v 2

which is the same equation with a smaller value of c. (ii) The equation a 4 - b4 = c2 has no solutions in integers with abc -:fa 0. (A similar argument works-the cases c even and c odd have to be considered separately.) Deduce from (ii) that no distinct Pythagorean triples can have two elements m common. 8.16 Find all solutions of y 2 = x(x 2 - 1) in Z/nZ for n = 5, 13 and 17 and hence verify the values given in section 8.4.4 for as, a13, a17. 8.17 For all integers n E [-50, 50], use the algorithm of section 8.4.3 to find the group of elements of finite order on the ellpitic curves y 2 = x 3 + nx and y 2 = x 3 +n. Then try curves y 2 = x 3 +ax+b, for instance with a, b E [-12, 12]. Can you make some conjectures, for instance about when the group might be trivial? (A computer will obviously help, but try not to use existing tables except for checking.)

References [AB) G. Almkvist and B. Berndt. Gauss, Landen, Ramanujan, the Arithmetic Mean, Ellipses, 7r and the Ladies' Diary. American Mathematical Monthly, 95:585-608, 1988. [Ah) L. V. Ahlfors. Complex Analysis. McGraw-Hill, second edition, 1966. [AMM) Editorial comment. Solution to problem 6672. American Mathematical Monthly, 100:803-806, 1993. [Apl) T. M. Apostol. Mathematical Analysis. Addison-Wesley, 1957. [Ap2) T. M. Apostol. Introduction to Analytic Number theory. Springer, 1976. [AS) M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover, 1965. [BB) J. M. Borwein and P. B. Borwein. Pi and the ACM. Wiley, 1987. [BSD) B. Birch and H. P. F. Swinnerton-Dyer. Notes on elliptic curves I & II. lour. Reine Angew. Math., 212, 218:7-25, 79-108, 1963, 1965. [Cal) J. W. S. Cassels. Lectures on Elliptic curves. Cambridge U. P., 1991. [Ca2) J. W. S. Cassels. The rational solutions of the diophantine equation y 2 = x 3 - d. Acta Math, 82:243-73, 1950. [Ch) K. Chandrasekharan. Elliptic Functions. Springer, 1985. [Coh) H. Cohen. A Course in Computational Algebraic Number Theory. Springer, 1993. [Cox) D. A. Cox. An introduction to Fermat's Last Theorem. American Mathematical Monthly, 101:3-14, 1994. [DEGM) R. K. Dodd, J.C. Eilbeck, J. D. Gibbon and H. C. Morris. Solitons and Nonlinear Wave equations. Academic Press, 1982. [Er) A. Erdelyi et al. Tables of Integral Transforms, Vols I €3 II. McGraw-Hill, 1954. [Fu) W. Fulton. Algebraic Curves. Benjamin/Cummings, 1969. [HC) A. Hurwitz and R. Courant. Allgemeine Funktionentheorie und elliptische Funktionen. Springer, fourth edition, 1964. [HW) G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Clarendon Press, Oxford, fifth edition, 1979. [IR) K. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory. Springer, 1982. [JS) G. A. Jones and D. Singerman. Complex Functions. Cambridge U. P., 1987. [Ko) N. Koblitz. Introduction to Elliptic Curves and Modular Forms. Springer, 1984. [Neh) Z. Nehari. Conformal Mapping. Dover, 1975. [Nev) E. H. Neville. Jacobian Elliptic Functions. Clarendon Press, Oxford, second edition, 1951.

212

REFERENCES

[PS] G. P6lya and G. Szego. Problems and Theorems in Analysis I €3 IL Springer, second edition, 1976, 1978. [Ra] H. Rademacher. Topics in Analytic Number Theory. Springer, 1973. [Re] R. Remmert. Theory of Complex Functions. Springer, 1991. [RG] H. Rademacher and E. Grosswald. Dedekind Sums. Math. Assoc. of America, 1972. [Ro] H. E. Rose. A Course in Number Theory. Clarendon Press, Oxford, 1988. [Sc] B. Schoeneberg. Elliptic Modular Functions. Springer, 1974. [ST] J. H. Silverman and J. Tate. Rational Points on Elliptic Curves. Springer, 1992. [SY] W. C. Sellar and R. J. Yeatman. 1066 and all that. Methuen, 1975. [Wal] P. L. Walker. An Introduction to Complex Analysis. Hilger, 1974. [Wa2] P. L. Walker. The Theory of Fourier Series and Integrals. Wiley 1986. [We] A. Weil. Elliptic functions according to Eisenstien and Kronecker. Springer, 1976. [WW] E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge U.P., fourth edition, 1927.

Index alternating series 19, 106 amplitude 139 analytic 1 analytic identity principle 10 anharmonic group 157 arithmetic-geometric mean 141 Bernoulli functions 60 Bernoulli numbers 23 Bernoulli polynomials 58 Beta integrals 52 circular functions 11 inverse 27 cnoidal waves 17 4 congruent 70 connected 2 convergent 3 absolutely 4 pointwise 5 uniform 5 unordered 4 convolution 63 copolar 111 cubic equations 168 Dedekind 17-function 161 differentiable 1 differential equations 79, 81, 114, 172 division points 169 double factorial 68 Eisenstein convention 3, 13 Eisenstein series 11, 13 elliptic curves 188 complex 188 conjecture of Birch and SwinnertonDyer 204 divisibility 190, 195 group structure 189 L-function of 204

over finite fields 203 rank 201 rational 195 real 191 rectangular 192 rhombic 192 singular 192 torsion 197 elliptic integrals 121, 124, 131 elliptic modular function 156 entire functions 2, 177 equianharmonic case 166 Euler polynomials 68 Euler-Maclaurin summation formula 55, 60 Euler's constant 4 7 Euler's integral 50 exponential function 20 Farey series 169 Fermat equation 207 Fourier coefficients 99 Fourier series 98, 115 Fourier transform 62 fundamental regions 153 fundamental theorem of algebra 41 g-summation 10 gamma functions 43 Gaussian sum 181 Glaisher's notation for elliptic functions 126 harmonic number 45 holomorphic 1 infinite product 7, 18 integration 8 Jacobi symbol 181, 185 Jacobi's triple product 92

214 Jacobian functions 105 in terms of theta functions 110 integral of 120 logarithm of 120 power series for 146 Kortweg de Vries equation 172 Landen transform 77, 148 lattices 70 Laurent series 7 Legendre symbol 183 Leibniz' series 32 lemniscate functions 14 7 lemniscatic case 166 Liouville's theorem 88 logarithms 23 holomorphic 40 principal 26 'memorable' formulae 128 modular angle 140 modular functions 151 modular groups 151 number theory 180 open mapping principle 41, 134 period parallelogram 70 period set 12 Poisson's formula 101 problem of inversion 125, 164 quasi-periodic 84 radius of convergence 6 region 3 Riemann zeta function 103 sequence 3 series 3 simple pendulum 122 spherical pendulum 149 square root 10 principal 26 soliton 174 star-shaped 9 Stirling's formula 56, 61 Taylor series 7

INDEX theorem addition 15, 77, 81, 84, 95, 113, 147 duplication 15, 45, 76, 80 of Feuter 200 of Mazur 197 of N agell and Lutz 197 of Picard 177 quasi-duplication 77 rotation 75, 80, 96, 109 theta functions 87 derivatives 114 sums of squares 93, 114 theta series 91 volume of the n-sphere 67 Wallis' product 32 waves 171 Weierstrass functions 80, 83 winding number 40 zeta functions 103