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combine/polylog

combine polylog functions

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

combine(f, polylog);

Parameters

f

-

any expression

Description

  

Expressions involving polylog are combined as follows

polylogn,x+−1npolylogn,1x=2k=1iquon,2lnxn2kpolylog2k,−1n2k!lnxnn!

polylogn,1y+−1npolylogn,y=2k=1iquon,2ln1yn2kpolylog2k,−1n2k!ln1ynn!

polyloga,z+polyloga,z=21apolyloga,z2

polylog2,z+polylog2,1z=π26lnzln1z

polylog2,w+polylog2,ww1={12ln1w2w112π22Iπlnw+Iπlnw112lnw121<w

polylog3,w+polylog3,1w+polylog3,ww1=polylog3,1+16π2ln1w12lnwln1w2+16ln1w3,forw<1

  

where n is an integer greater than one, a, w, x, y, and z are variables assumed to be complex, where 1x, and y<1.

Examples

combinepolyloga&comma;x&plus;polyloga&comma;x&comma;polylog

21apolyloga&comma;x2

(1)

assume1<x

combinepolylog2&comma;x&plus;polylog2&comma;xx1&comma;polylog

π222Iπlnx~+Iπlnx~1lnx~122

(2)

assumex&comma;RealRange1&comma;1

combinepolylog4&comma;x&plus;polylog4&comma;1x&comma;polylog

ln1x~2π2127π4360ln1x~424

(3)

assumex<1

combinepolylog3&comma;x&plus;polylog3&comma;1x&plus;polylog3&comma;xx1&comma;polylog

ζ3+π2ln1x~6lnx~ln1x~22+ln1x~36

(4)

See Also

polylog