Elliptic Functions [Hardcover]

A beginning graduate level treatment of elliptic functions with a huge array of examples, first published in 2006.The opening chapters of this 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?' Later chapters present a more conventional approach to the Weierstrass functions and elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic, dynamics, and probability and statistics, are discussed.The opening chapters of this 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?' Later chapters present a more conventional approach to the Weierstrass functions and elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic, dynamics, and probability and statistics, are discussed.In its first six chapters, this text presents the basic ideas and properties of the Jacobi elliptic functions as a historical essay. 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 the reader is introduced to the richly varied applications of the elliptic and related functions.1. The 'simple' pendulum; 2. Jacobian elliptic functions of a complex variable; 3. General properties of elliptic functions;l38