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# PolyLog

 PolyLoggives the polylogarithm function . PolyLoggives the Nielsen generalized polylogarithm function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• .
• .
• PolyLog has a branch cut discontinuity in the complex plane running from 1 to .
• For certain special arguments, PolyLog automatically evaluates to exact values.
• PolyLog can be evaluated to arbitrary numerical precision.
• PolyLog automatically threads over lists.
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 Scope   (7)
Simple exact values are generated automatically:
Evaluate with large order and argument:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Evaluate for complex order and arguments:
PolyLog threads element-wise over lists:
Infinite arguments give symbolic results:
PolyLog can be applied to power series:
Evaluate derivatives exactly:
Series expansion at branch cuts:
Series expansion at infinity:
Give the result for an arbitrary symbolic direction:
Special cases:
Series expansion:
 Applications   (4)
Plot of the absolute value of the dilogarithm function in the complex plane:
Calculate integrals over Fermi-Dirac distributions:
Volume of a hyperbolic dodecahedron with vertices 0, 1, , (subject to ):
Plot the volume as a function of the vertex :
Mahler measure of the trivariate polynomial as a function of :
Plot the Mahler measure:
Use FullSimplify to simplify polylogarithms:
Use FunctionExpand to expand polylogarithms:
Numerically find a root of a transcendental equation:
Integration:
Generate from integrals and sums:
PolyLog appears in special cases of various mathematical functions:
Large orders can give results too large to be computed explicitly:
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