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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 \[ L_2(z) = - \int_0^z \frac{\ln{(1-t)}}{t}\,dt\,, \quad z \in \mathbb{C} \setminus (-\infty,0]\,, \] 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.

MSC:

33B30 Higher logarithm functions
33-02 Research exposition (monographs, survey articles) pertaining to special functions
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Digital Library of Mathematical Functions:

(25.12.1) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.2) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.3) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.4) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.5) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.6) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.7) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.8) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.12.9) ‣ §25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
Integral Representation ‣ §25.12(ii) Polylogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions