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Logarithmic functions --- Logarithms --- Transcendental functions

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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.

Elliptic functions --- Elliptic functions. --- Transcendental functions. --- Functions, Transcendental --- Functions, Special --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic

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Combinatorial identities --- Generating functions --- Hypergeometric functions --- Partitions (Mathematics) --- Number theory --- Functions, Hypergeometric --- Transcendental functions --- Hypergeometric series --- Functions, Generating --- Combinatorial analysis --- Identities, Combinatorial

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#WBIB:dd.Lic.L.De Busschere --- Bessel functions --- Cylindrical harmonics --- Transcendental functions --- Bessel polynomials --- Harmonic analysis --- Harmonic functions --- Fonctions speciales --- Fonctions de bessel

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A comprehensive treatment of elliptic functions is linked by these notes to a study of their application to elliptic curves. This approach provides geometers with the opportunity to acquaint themselves with aspects of their subject virtually ignored by other texts. The exposition is clear and logically carries themes from earlier through to later topics. This enthusiastic work of scholarship is made complete with the inclusion of some interesting historical details and a very comprehensive bibliography.

Elliptic functions. --- Curves, Elliptic. --- Elliptic curves --- Curves, Algebraic --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Curves, Algebraic.