Topic #9 ------------- OP-SF NET 5.2 ------------ March 15, 1998 ~~~~~~~~~~~~~ From: Wolfram KoepfSubject: 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 work. 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) and 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) Contents: 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 functions 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. References Index 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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