PolyLog Function

a

0.5

b

25.

b

0

0.86

f

Re

Im

Abs

Arg

view

2D

density

3D

The polylogarithm function (or Jonquière's function)

Li

s

(z)

of index

s

and argument

z

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(

Li

a+bi

(x+

b

0

i))

, and the 3D plot shows

(x,y)⟶f(

Li

a+bi

(x+yi))

.