PolyLog Function

0.5

25.

0.86

Abs

Arg

view

density

The polylogarithm function (or Jonquière's function)

(z)

of index

and argument

is a special function, defined in the complex plane for

|z|<1

and by analytic continuation otherwise. It can be plotted for complex values

s=a+bi

; for example, along the celebrated critical line

s=1/2+bi

for Riemann's zeta function[1]. The polylogarithm function appears in the Fermi–Dirac and Bose–Einstein distributions and also in quantum electrodynamics calculations for Feynman diagrams. The 2D plot shows the function

x⟶f(

a+bi

(x+

i))

, and the 3D plot shows

(x,y)⟶f(

a+bi

(x+yi))

Details

The polylogarithm function is defined as

(z)=

∞

∑

k=1

For

s=1

, it is equivalent to the natural logarithm,

(z)=ln(1-z)

. For

s=2

and

s=3

, it is called the dilogarithm and the trilogarithm; the integral of a polylogarithm is itself a polylogarithm

s+1

(z)=

∫

(t)

References

[1] L. Vepstas. "Polylogarithm, The Movie." (Nov 20, 2014) linas.org/art-gallery/polylog/polylog.html.

[2] T. M. Apostol. "Zeta and Related Functions." NIST Digital Library of Mathematical Functions, Version 1.0.9, Release date 2014-08-29. dlmf.nist.gov/25.12.

[3] Souichiro-Ikebe. "Polylogarithm Function." (Dec 4, 2015) Graphics Library of Special Functions (in Japanese). http://math-functions-1.watson.jp/sub1_spec_040.html.

External Links

Dilogarithm (Wolfram MathWorld)

Trilogarithm (Wolfram MathWorld)

Polylogarithm (Wolfram MathWorld)

PolyLog (The Wolfram Functions Site)

Permanent Citation

Enrique Zeleny

"PolyLog Function"
http://demonstrations.wolfram.com/PolyLogFunction/
Wolfram Demonstrations Project
Published: November 24, 2014