SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #9   -------------   OP-SF NET 5.2  ------------  March 15, 1998
From: Wolfram Koepf 
Subject: Review of book on Elliptic Functions

(This item appeared in our Activity Group's Newsletter, vol 8, no 2,
February 1998, p. 15)

Elliptic Functions. A Constructive Approach

By Peter L. Walker

Wiley, Chichester-New York-Brisbane-Toronto--Singapore, 214 pp., 1996,
ISBN 0-471-96531-6

This is a very remarkable book in which the theory of elliptic, i.e.
doubly-periodic, functions is completely developed by direct manipulations
of series, products and integrals. Hence the author takes a rather
algebraical point of view which is primarily stimulated by Eisenstein's

Furthermore, to give a unified treatment, the author develops the theory
of circular and related functions (like gamma) by the same method.

Indeed, the starting point for the circular and related functions are the
families of series (k in Z, k > 0)

             sum {n = -infty to infty} (x+n)^(-k)                     (1)


               sum {n = 0 to infty} (x+n)^(-k),                       (2)

whereas the corresponding starting point for the elliptic functions is
given by the family of double series (k in Z, k > 0, Im tau > 0)

 sum{n=-infty to infty} sum {m=-infty to infty} (x + m +n tau)^(-k).  (3)


0. Preliminaries:  Here a short introduction to series, products, and
integrals is given. 

1. Circular Functions:  Starting with (1), the cotangent function, pi and
finally the other circular functions and their properties are developed. 

2. Gamma and Related Functions:  Using (2)  instead of (1)  yields the
gamma and related functions. 

3. Basic Elliptic Functions: Starting from (3), the Weierstrass elliptic
functions are developed.

4. Theta Functions: Series similar to (3), e.g. with alternating sign, for
k = 1 lead to the Jacobian theta functions. 

5. Jacobian Functions:  The Jacobian elliptic functions are ratios of
Jacobian theta functions, and come next. 

6. Elliptic Integrals: Elliptic integrals as inverses of elliptic
are discussed now.

7. Modular Functions: Here, modular functions are treated. These are
functions depending on the period ratio tau. 

8. Applications:  The use of elliptic functions in connection with waves,
number theory and elliptic curves are discussed. 



I am very impressed by the author's treatment. It is rather striking how

  1. the theories of the circular and the elliptic functions can be
      developed in much the same manner;
  2. these developments can be done in such an algebraic way.

I recommend this book warmly to everybody who is interested in looking
"behind the scenes" of elliptic functions.

Wolfram Koepf

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