# The dilogarithm function for complex argument

@article{Maximon2003TheDF, title={The dilogarithm function for complex argument}, author={Leonard C. Maximon}, journal={Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences}, year={2003}, volume={459}, pages={2807 - 2819} }

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…

## 81 Citations

### New integral representations of the polylogarithm function

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2006

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the…

### The Polylogarithm Function in Julia

- Computer ScienceArXiv
- 2020

This paper presents an algorithm for calculating polylogarithms for both complex parameter and argument and evaluates it thoroughly in comparison to the arbitrary precision implementation in Mathematica.

### New definite integrals and a two-term dilogarithm identity from a hyperbolic change of variables

- Mathematics
- 2010

### Polylogarithms, functional equations and more: The elusive essays of William Spence (1777–1815)

- Mathematics
- 2013

### Harmonic Sums, Polylogarithms, Special Numbers, and their Generalizations

- MathematicsArXiv
- 2013

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and…

### Some simple proofs of Lima's two-term dilogarithm identity

- MathematicsIrish Mathematical Society Bulletin
- 2022

. Recently, Lima found a remarkable two-term dilogarithm identity whose proof was based on a hyperbolic form of a proof for the Basel problem given by Beukers, Kolk, and Calabi. A number of simple…

### The polylogarithm and the Lambert W functions in thermoelectrics

- Mathematics
- 2011

In this work, we determine the conditions for the extremum of the figure of merit, θ2, in a degenerate semiconductor for thermoelectric (TE) applications. We study the variation of the function θ2…

### Perturbative Quantum Field Theory Meets Number Theory

- Mathematics
- 2014

Feynman amplitudes are being expressed in terms of a well structured family of special functions and a denumerable set of numbers—periods, studied by algebraic geometers and number theorists. The…

### On Some Integral Representation Of $\zeta(n)$ Involving Nielsen's Generalized Polylogarithms And The Related Partition Problem

- Mathematics
- 2021

In this paper, we study a family of single variable integral representations for some products of ζ(2n + 1), where ζ(z) is Riemann zeta function and n is positive integer. Such representation…

## References

SHOWING 1-10 OF 52 REFERENCES

### Dilogarithm identities

- Mathematics
- 1994

We study the dilogarithm identities from algebraic, analytic, asymptotic, $K$-theoretic, combinatorial and representation-theoretic points of view. We prove that a lot of dilogarithm identities…

### Polylogarithms and Riemann's ζ function

- Physics
- 1997

Riemann’s z function perhaps first appeared in statistical mechanics in 1900 in Planck’s theory of the blackbody radiation and then in 1912 in Debye’s theory of the specific heats of solids @1#.…

### On Nielsen's generalized polylogarithms and their numerical calculation

- Mathematics
- 1970

The generalized polylogarithms of Nielsen are studied, in particular their functional relations. New integral expressions are obtained, and relations for function values of particular arguments are…

### Quantum Electrodynamics

- PhysicsNature
- 1947

THE subject of quantum electrodynamics is extremely difficult, even for the case of a single electron. The usual method of solving the corresponding wave equation leads to divergent integrals. To…

### Handbook of Mathematical Functions with Formulas, Graphs,

- Mathematics
- 1971

The DLMF, also being published as a book, is the successor to Abramowitz and Stegun's NBS Handbook of Mathematical Functions with Formulas, Graphs. Handbook of Mathematical Functions has 24 ratings…

### Polylogarithmic analysis of chemical potential and fluctuations in a D‐dimensional free Fermi gas at low temperatures

- Physics
- 1995

The chemical potential and fluctuations in number of particles in a D‐dimensional free Fermi gas at low temperatures are obtained by means of polylogarithms. This idea is extended to show that the…

### Nielsen's generalized polylogarithms

- Mathematics
- 1986

Properties (in particular functional relations and special values) of the functions \[\begin{gathered} ( - 1)^{n + p - 1} (n - 1)!p!S_{n,p} (z) = \int_0^1 {\log ^{n - 1} t\log ^p (1 - zt)\frac{{dt}}…

### Fourth-order radiative corrections to electron-photon vertex and the Lamb-shift value

- Physics
- 1971

SummaryThe slope of the charge form factor of the electron is analytically evaluated in fourth-order perturbation theory of QED and found to bem2F′1(0)=(α/π)2[−4819/5184−(49/72)ζ(2)+3ζ(2) log…