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The dilogarithm function for complex argument. (English) Zbl 1050.33002
The Euler dilogarithm, often referred to as the Spence function, is defined by $$ L_2(z) = - \int_0^z \frac{\ln{(1-t)}}{t}\,dt\,, \quad z \in \Bbb{C} \setminus (-\infty,0]\,, $$ where $\ln$ is the principal branch of the logarithm. This article is an exposition of the basic properties of the dilogarithm. These include analytic continuation, integral representations, transformation formulae, series expansions, functional relations, numerical values for special arguments, relations to hypergeometric and generalized hypergeometric functions and relations to the inverse tangent integral and Clausen’s integral. The author also gives a brief summary of generalizations of the dilogarithm, namely polylogarithms, Nielsen’s generalized polylogarithms, Jonquière’s function and Lerch’s function. The article closes with some historical notes and references to applications in physics and mathematics.

33B30Higher logarithm functions
33-02Research monographs (special functions)
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