Royal Society Publishing

The dilogarithm function for complex argument

Leonard C Maximon

Abstract

This paper summarizes the basic properties of the Euler dilogarithm function, often referred to as the Spence function. These include integral representations, series expansions, linear and quadratic transformations, functional relations, numerical values for special arguments and relations to the hypergeometric and generalized hypergeometric function. The basic properties of the two functions closely related to the dilogarithm (the inverse tangent integral and Clausen's integral) are also included. A brief summary of the defining equations and properties for the frequently used generalizations of the dilogarithm (polylogarithm, Nielsen's generalized polylogarithm, Jonquière's function, Lerch's function) is also given. A résumé of the earliest articles that consider the integral defining this function, from the late seventeenth century to the early nineteenth century, is presented. Critical references to details concerning these functions and their applications in physics and mathematics are listed.

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