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Zbl 1050.33002
Maximon, Leonard C.
The dilogarithm function for complex argument. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 459, No. 2039, 2807-2819 (2003). ISSN 1364-5021; ISSN 1471-2946/e

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.
[Rainer Brück (Dortmund)]

MSC 2000:: *33B30 Higher logarithm functions
33-02 Research monographs (special functions)

Keywords: Euler dilogarithm; Spence function; Debye function; Jonquère's function; polylogarithms; Clausen's integral

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