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The Dilogarithm Function for Complex Argument



L C. Maximon


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 specialarguments, and its relation 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 definingequations and properties for the frequently utilized generalizations of the dilogarithm (polylogarithm, Nielsen's generalized polylogarithm, Lerch's transcendent) is also given. Critical references to details concerning these functions and their applications are listed.
Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences


Clausen's integral, dilogarithm function, Euler dilogarithm, inverse tangent integral, Nielsen's generalized polylogarithm, polylogarithm, spence function


Maximon, L. (2003), The Dilogarithm Function for Complex Argument, Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, [online], (Accessed June 23, 2024)


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Created November 8, 2003, Updated June 2, 2021