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.
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The book provides the reader with some precious concrete scientific material , which is very useful in number theory. The book collects all facts about the elliptic functions which are the backbone of the number theory.
This is a very BAD good book. The way the material is presented is great but the number of typographical errors makes it unreadable especially by students. The first typo is page xi, and you can find pages with more than 4 typographical errors (e.g. p. 46 to 48), I stopped counting them after reaching one hundred!
Mathematical typography is bad, even inconsistent, (Jacobi functions, sn, cn... are in italic instead of regular but not always (p.60, one sn is OK, so why not all) and sometimes you find dn written with d straight and n in italic, p. 234,239!). Very often figures are wrong, one is missing (p. 255).
I have never bought a book with so many mistakes, this book is not worth the money I spent on it.
Jacobian elliptic functions of a complex variable
General properties of elliptic functions
The Jacobian elliptic functions for complex k
Introduction to transformation theory
The Weierstrass elliptic functions
Applications of elliptic functions in geometry
Applications in mechanics statistics and other topics
An application of elliptic functions in algebra solution of the general quintic equation 276