Elliptic Functions

A beginning graduate level treatment of elliptic functions with a huge array of examples, first published in 2006.

J. V. Armitage (Author), W. F. Eberlein (Author)

9780521785631, Cambridge University Press

Paperback, published 28 September 2006

402 pages, 25 b/w illus. 5 tables
22.9 x 15.2 x 2.3 cm, 0.55 kg

'This solid text is a good place to start when working with elliptic functions and it is the sort of book that you will keep coming back to as reference text.' Mathematics Today

In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.

1. The 'simple' pendulum
2. Jacobian elliptic functions of a complex variable
3. General properties of elliptic functions
4. Theta functions
5. The Jacobian elliptic functions for complex k
6. Introduction to transformation theory
7. The Weierstrass elliptic functions
8. Elliptic integrals
9. Applications of elliptic functions in geometry
10. An application of elliptic functions in algebra solution of the general quintic equation
11. An arithmetic application of elliptic functions
12. Applications in mechanics and statistics and other topics
Appendix
Bibliography.

Subject Areas: Combinatorics & graph theory [PBV], Probability & statistics [PBT], Calculus & mathematical analysis [PBK], Number theory [PBH]