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1985 | Buch

Periods of meromorphic functions Kapitel lesen Erstes Kapitel lesen

Elliptic Functions

verfasst von: Komaravolu Chandrasekharan

Verlag: Springer Berlin Heidelberg

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

Inhaltsverzeichnis

Frontmatter
Chapter I. Periods of meromorphic functions
Abstract
We assume as known the fundamentals of complex analysis, including the basic properties of holomorphic and of meromorphic functions in the complex plane. The meromorphic functions defined on an open, connected, set in the complex plane form a field. Unless otherwise qualified, a meromorphic function is supposed to mean a function meromorphic in the whole complex plane.
Komaravolu Chandrasekharan
Chapter II. General properties of elliptic functions
Abstract
Given an elliptic function f, let (ω1 ω2) be a pair of basic periods for its period-lattice {mω1 + nω2}, where m, n = 0, ±1, ±2,....
Komaravolu Chandrasekharan
Chapter III. Weierstrass’s elliptic function ℘(z)
Komaravolu Chandrasekharan
Chapter IV. The zeta-function and the sigma-function of Weierstrass
Abstract
Weierstrass’s ζ-function is a meromorphic function, which has simple poles, with residues equal to one, at all points which correspond to the periods of Weierstrass’s ℘-function. It is not elliptic. But every elliptic function can be expressed in terms of ζ and its derivatives; in fact ζ’(z)= -℘(z).
Komaravolu Chandrasekharan
Chapter V. The theta-functions
Abstract
The expansions, in infinite series, of the functions ℘(u), ζ(u), and σ(u), which we have so far considered, are not best suited to numerical computation. It is of advantage therefore to introduce another function, denoted by θ(υ, τ), which has a rapidly convergent expansion in infinite series, and which is directly connected with the σ-function of Weierstrass.
Komaravolu Chandrasekharan
Chapter VI. The modular function J(τ)
Abstract
A complex-valued function F(z) of the complex variable z is said to be a modular function, if it is meromorphic for lm z>0, and F(Mz) = F(z) for all transformations M belonging to the modular group (defined in § 5 of Chapter I), or for all M belonging to a sub-group of the modular group of finite index.
Komaravolu Chandrasekharan
Chapter VII. The Jacobian elliptic functions and the modular functionλ(τ)
Abstract
Letτbe a complex number, with Im τ>0. Letω1be defined as a function ofτby the relation
$$ {\omega _{1}} = \pi \theta _{3}^{2}(0,\tau ) = \pi {(1 + 1q + 2{q^{4}} + ...)^{2}},q = {e^{{\pi i\tau }}},\operatorname{Im} \tau >0$$
(1.1)
and let\( {\omega _{2}} = {\omega _{1}} \cdot \tau \). The Jacobian elliptic function snuis a doubly-periodic, meromorphic function ofu, with\( \left( {{\omega _{1}},{\omega _{2}}} \right) \)as a pair of basic periods, with two simple poles in each period-parallelogram, the sum of the residues at those poles being zero. It satisfies the differential equation
$$ {\left( {\frac{{dy}}{{du}}} \right)^{2}} = \left( {1 - {y^{2}}} \right)\left( {1 - {k^{2}}{y^{2}}} \right),\quad y = snu $$
(1.2)
, where
$$ {k^{2}} = {k^{2}}\left( \tau \right) = \frac{{\theta _{1}^{4}\left( {0,\tau } \right)}}{{\theta _{3}^{4}\left( {0,\tau } \right)}} $$
(1.3)
.
Komaravolu Chandrasekharan
Chapter VIII. Dedekind’s η-function and Euler’s theorem on pentagonal numbers
Komaravolu Chandrasekharan
Chapter IX. The law of quadratic reciprocity
Abstract
As a limiting case of the transformation formula connecting the theta-function θ 3 (υ, τ) with \( % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeWaaeWaa8aa % baWdbiaadAhacaGGSaGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aaba % Wdbiabes8a0baaaiaawIcacaGLPaaacaGGSaaaaa!40B9! {\theta _3}\left( {v, - \frac{1}{\tau }} \right), \), we shall prove a transformation formula for exponential sums (Theorem 1), which yields, as a special case, a reciprocity formula for generalized Gaussian sums (Corollary 2) which, in turn, enables us not only to evaluate Gaussian sums but to prove the law of quadratic reciprocity.
Komaravolu Chandrasekharan
Chapter X. The representation of a number as a sum of four squares
Abstract
We have seen in Chapter VIII that the identity
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH3oaAdaqadaWdaeaapeGaamOEaaGaayjkaiaawMcaaiabg2da % 9iaadwgapaWaaWbaaSqabeaapeGaeqiWdaNaamyAaiaadQhacaGGVa % GaaGymaiaaikdaaaGccqaH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqa % baGcpeWaaeWaa8aabaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qaca % aIYaaaaiabgUcaRmaalaaapaqaa8qacaWG6baapaqaa8qacaaIYaaa % aiaacYcacaaIZaGaamOEaaGaayjkaiaawMcaaiaacYcaciGGjbGaai % yBaiaadQhacqGH+aGpcaaIWaaaaa!535D! \eta \left( z \right) = {e^{\pi iz/12}}{\theta _3}\left( {\frac{1}{2} + \frac{z}{2},3z} \right),\operatorname{Im} z > 0 $$
(1.1)
, which connects Dedekind’s η-function with the theta-function 6 3 implies Euler’s theorem on pentagonal numbers. That was proved by analytical methods in two different ways. The first consisted in representing θ3(υ, z), initially defined by an infinite series, as an infinite product, and identifying the defining product of η(z) with that which results from the right-hand side of (1.1). The second consisted in combining the transformation formula for θ3(υ, z), namely
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeWaaeWaa8aa % baWdbiaaicdacaGGSaGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aaba % WdbiaaikdaaaaacaGLOaGaayzkaaGaeyypa0ZaaOaaa8aabaWdbmaa % laaapaqaa8qacaWG6baapaqaa8qacaWGPbaaaiaac6caaSqabaGccq % aH4oqCpaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeWaaeWaa8aabaWd % biaaicdacaGGSaGaamOEaaGaayjkaiaawMcaaiaacYcaciGGjbGaai % yBaiaadQhacqGH+aGpcaaIWaaaaa!4F1F! {\theta _3}\left( {0, - \frac{1}{2}} \right) = \sqrt {\frac{z}{i}.} {\theta _3}\left( {0,z} \right),\operatorname{Im} z > 0 $$
(1.2)
, with the functional equation of η(z), namely
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH3oaAdaqadaWdaeaapeGaeyOeI0YaaSaaa8aabaWdbiaaigda % a8aabaWdbiaaikdaaaaacaGLOaGaayzkaaGaeyypa0ZaaOaaa8aaba % Wdbmaalaaapaqaa8qacaWG6baapaqaa8qacaWGPbaaaiabeE7aOnaa % bmaapaqaa8qacaWG6baacaGLOaGaayzkaaaaleqaaOGaaiilaiGacM % eacaGGTbGaamOEaiabg6da+iaaicdaaaa!4923! \eta \left( { - \frac{1}{2}} \right) = \sqrt {\frac{z}{i}\eta \left( z \right)} ,\operatorname{Im} z > 0 $$
(1.3)
, (1.3) so as to construct a modular function which vanishes identically in the upper half-plane Im z>0, and thereby yields (1.1).
Komaravolu Chandrasekharan
Chapter XI. The representation of a number by a quadratic form
Abstract
The arithmetical function r(n), defined as the number of representations of a positive integer n by the form \( % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGUbWdamaaDaaaleaapeGaaGymaaWdaeaapeGaaGOmaaaakiab % gUcaRiaad6gapaWaa0baaSqaa8qacaaIYaaapaqaa8qacaaIYaaaaO % Gaey4kaSIaamOBa8aadaqhaaWcbaWdbiaaiodaa8aabaWdbiaaikda % aaGccqGHRaWkcaWGUbWdamaaDaaaleaapeGaaGinaaWdaeaapeGaaG % Omaaaaaaa!4432! n_1^2 + n_2^2 + n_3^2 + n_4^2 \) where the (n k ) are integers, is related to the fourth power of the theta-function θ3(0, z), which is defined by the theta-series
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaqa % aaaaaaaaWdbiaadghapaWaaWbaaSqabeaapeGaamOBa8aadaqhaaad % baWdbiaaigdaa8aabaWdbiaaikdaaaWccqGHRaWkcaWGUbWdamaaDa % aameaapeGaaGOmaaWdaeaapeGaaGOmaaaaliabgUcaRiaad6gapaWa % a0baaWqaa8qacaaIZaaapaqaa8qacaaIYaaaaSGaey4kaSIaamOBa8 % aadaqhaaadbaWdbiaaisdaa8aabaWdbiaaikdaaaaaaOGaaiilaiaa % ywW7caWGXbGaeyypa0ZaaWbaaSqabeaacaWGLbGaeqiWdaNaamyAai % aadQhaaaGccaGGSaGaaGzbVlGacMeacaGGTbGaamOEaiabg6da+iaa % icdacaGGSaaal8aabaWdbiaad6gapaWaaSbaaWqaa8qacaaIXaaapa % qabaWcpeGaaiilaiaad6gapaWaaSbaaWqaa8qacaaIYaaapaqabaWc % peGaaiilaiaad6gapaWaaSbaaWqaa8qacaaIZaaapaqabaWcpeGaai % ilaiaad6gapaWaaSbaaWqaa8qacaaI0aaapaqabaWcpeGaeyypa0Ja % eyOeI0IaeyOhIukapaqaaiabg6HiLcqdcqGHris5aOGaafiiaaaa!68D1! \sum\limits_{{n_1},{n_2},{n_3},{n_4} = - \infty }^\infty {{q^{n_1^2 + n_2^2 + n_3^2 + n_4^2}},\quad q{ = ^{e\pi iz}},\quad \operatorname{Im} z > 0,} {\text{ }} $$
, by the formula
$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaH4oqCpaWaa0baaSqaa8qacaaIZaaapaqaa8qacaaI0aaaaOWa % aeWaa8aabaWdbiaaicdacaGGSaGaamOEaaGaayjkaiaawMcaaiabg2 % da9maaqahapaqaa8qacaWGYbWaaeWaa8aabaWdbiaad6gaaiaawIca % caGLPaaacaWGXbWdamaaCaaaleqabaWdbiaad6gaaaGccaGGSaGaam % OCamaabmaapaqaa8qacaaIWaaacaGLOaGaayzkaaGaeyypa0JaaGym % aiaacYcacaWGXbGaeyypa0Jaamyza8aadaahaaWcbeqaa8qacqaHap % aCcaWGPbGaamOEaaaakiaacYcaciGGjbGaaiyBaiaadQhacqGH+aGp % caaIWaaal8aabaWdbiaad6gacqGH9aqpcaaIWaaapaqaa8qacqGHEi % sPa0GaeyyeIuoaaaa!5DF6! \theta _3^4\left( {0,z} \right) = \sum\limits_{n = 0}^\infty {r\left( n \right){q^n},r\left( 0 \right) = 1,q = {e^{\pi iz}},\operatorname{Im} z > 0} $$
(cf. (2.1), Ch. X) which enabled us to determine r(n) in terms of the divisors of n.We now consider the more general problem of finding the number of representations of a positive interger by a positive-definite quadratic form, by constructing more general theta-series, and studying their behaviour under modular transformations.
Komaravolu Chandrasekharan
Backmatter
Metadaten
Titel
Elliptic Functions
verfasst von
Komaravolu Chandrasekharan
Copyright-Jahr
1985
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-52244-4
Print ISBN
978-3-642-52246-8
DOI
https://doi.org/10.1007/978-3-642-52244-4