Elliptic Functions

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.

• The emphasis on applications makes the book suitable for a lecture course in which students are invited to develop ideas as a basis for essays or projects
• Applications are referred to throughout and the book concludes with significant applications in algebra, arithmetic, classical geometry and applied mathematics
• A new look at the Jacobi functions based on a rigorous treatment of the differential equation for the simple pendulum and the inversion of integrals