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Elliptic functions

Author: K Chandrasekharan
Publisher: Berlin ; New York : Springer-Verlag, ©1985.
Series: Grundlehren der mathematischen Wissenschaften, 281.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

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  Read more...

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Additional Physical Format: Online version:
Chandrasekharan, K. (Komaravolu), 1920-
Elliptic functions.
Berlin ; New York : Springer-Verlag, ©1985
(OCoLC)624452531
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: K Chandrasekharan
ISBN: 0387152954 9780387152950 3540152954 9783540152958
OCLC Number: 12053023
Language Note: Based on lectures given at the Swiss Federal Institute of Technology, Zürich, during the summer semester of 1982.
Notes: Based on lectures given at the Swiss Federal Institute of Technology, Zürich, during the summer semester of 1982.
Description: xi, 189 pages : illustrations ; 24 cm.
Contents: I. Periods of meromorphic functions --
II. General properties of elliptic functions --
III. Weierstrass's elliptic function p(z) --
IV. The zeta-function and the sigma-function of Weierstrass --
V. The theta-functions --
x. Table of contents --
VI. The modular function J(.) --
VII. The Jacobian elliptic functions and the modular function A(.) --
VIII. Dedekind's 1]-function and Euler's theorem on pentagonal numbers --
IX. The law of quadratic reciprocity --
X. The representation of a number as a sum of four squares --
XI. The representation of a number by a quadratic form. I. Periods of meromorphic functions --
1. Meromorphic functions --
2. Periodic meromorphic functions --
3. Jacobi's lemma --
4. Elliptic functions --
5. The modular group and modular functions --
II. General properties of elliptic functions --
1. The period parallelogram --
2. Elementary properties of elliptic functions --
III. Weierstrass's elliptic function p(z) --
1. The convergence of a double series --
2. The elliptic function p(z) --
3. The differential equation associated with p(z) --
4. The addition-theorem --
5. The generation of elliptic functions --
Appendix I. The cubic equation : --
Appendix II. The biquadratic equation : --
IV. The zeta-function and the sigma-function of Weierstrass --
1. The function ((z) --
2. The function a(z) --
3. An expression for elliptic functions --
V. The theta-functions --
1. The function O(v, r) --
2. The four sigma-functions --
3. The four theta-functions --
4. The differential equation --
x. Table of contents --
5. Jacobi's formula for ()f (0, .) --
6. The infinite products for the theta-functions --
7. Theta-functions as solutions of functional equations --
8. The transformation formula connecting ()iv, .) and ()3(V, -f) --
VI. The modular function J(.) --
1. Definition of J(.) --
2. The functions g2(.) and gi.) --
3. Expansion of the function J(.) and the connexion with theta-functions --
4. The function J(.) in a fundamental domain of the modular group --
5. Relations between the periods and the invariants of 8O(U) --
6. Elliptic integrals of the first kind --
VII. The Jacobian elliptic functions and the modular function A(.) --
1. The functions sn u, en u, dn U of Jacobi --
2. Definition by theta-functions --
3. Connexion with the sigma-functions --
4. The differential equation --
5. Infinite products for the Jacobian elliptic functions --
6. Addition-theorems for sn u, en u, dn U --
7. The modular function A(.) --
8. Mapping properties of A(.) and Picard's theorem --
VIII. Dedekind's 1]-function and Euler's theorem on pentagonal numbers --
1. Connexion with the invariants of the so-function and with the theta-functions --
2. Euler's theorem and Jacobi's proof --
3. The transformation formula connecting 1](z) and 1]( -f) --
4. Siegel's proof of Theorem 1 --
5. Connexion between 1](z) and the modular functions J(z), A(Z) --
IX. The law of quadratic reciprocity --
1. Reciprocity of generalized Gaussian sums --
2. Quadratic residues --
3. The law of quadratic reciprocity --
X. The representation of a number as a sum of four squares --
1. The theorems of Lagrange and of Jacobi --
2. Proof of Jacobi's theorem by means of theta-functions --
3. Siegel's proof of Jacobi's theorem --
XI. The representation of a number by a quadratic form --
1. Positive-definite quadratic forms --
2. Multiple theta-series and quadratic forms --
3. Theta-functions associated to positive-definite forms --
4. Representation of an even integer by a positive-definite form.
Series Title: Grundlehren der mathematischen Wissenschaften, 281.
Responsibility: K. Chandrasekharan.

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