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polylog

general polylogarithm function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

polylog(a, z)

Parameters

a

-

expression

z

-

expression

Description

• 

The polylogarithm of index a at the point z is defined by

polyloga,z=n=1znna

  

if z<1 and by analytic continuation otherwise.  The index a can be any complex number.  If a1, the point z&equals;1 is a singularity.

• 

For all indices a, the point z&equals;1 is a branch point for all branches, and in Maple, the branch cut is taken to be the interval (1,).  For the branches other than the principal branch (which is given on the unit disk by the series above, and hence is analytic at 0), the point z&equals;0 is also a branch point, and the branch cut is taken to be the negative real axis.  The formula for a particular branch can be determined with the following rules:

  

Each time the branch cut (1,) is crossed in the counterclockwise direction, subtract 2Iπlnza1Γa. Add this quantity if the branch cut is crossed in the clockwise direction.

  

Each time the branch cut (,0) is crossed in the counterclockwise direction, add 2Iπ to each lnz term in the current formula.  Subtract this quantity if the branch cut is crossed in the clockwise direction.

  

For example, if one traverses a path which starts at z&equals;12, goes clockwise around z&equals;1, then counterclockwise around z&equals;0, then clockwise around z&equals;1 again to return at z&equals;12, the formula for the branch of polylog thus obtained would be

Liaz+2Iπlnza1+lnz+2Iπa1Γa

  

where polylog(a, z) indicates the principal branch and lnz means the principal branch of the logarithm.

• 

Maple only evaluates the principal branch.

• 

Maple's dilog function is related to polylog by the relation dilogz&equals;polylog2&comma;1z.

Examples

polyloga&comma;0

0

(1)

polylog2&comma;1

16&pi;2

(2)

polylog3&comma;1

&zeta;3

(3)

polylog2&comma;I

148&pi;2&plus;ICatalan

(4)

xpolyloga&comma;x

polyloga1&comma;xx

(5)

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

21apolyloga&comma;x2

(6)

polylog4&comma;x&plus;polylog4&comma;1x

polylog4&comma;x&plus;polylog4&comma;1x

(7)

combine&comma;polylogassuming1<x

112lnx2&pi;27360&pi;4124lnx4

(8)

combine&comma;polylogassumingx::RealRange1&comma;1

112ln1x2&pi;27360&pi;4124ln1x4

(9)

polyloga&comma;z5

polyloga&comma;z5

(10)

expand

155apolyloga&comma;12&sol;5z&plus;155apolyloga&comma;14&sol;5z&plus;155apolyloga&comma;z11&sol;5&plus;155apolyloga&comma;z13&sol;5&plus;155apolyloga&comma;z

(11)

x&apos;x&apos;&colon;

polylog1&comma;x

ln1x

(12)

polylog2&comma;13

polylog2&comma;13

(13)

evalf

0.3662132299

(14)

polylog3.7&plus;2.2I&comma;1.5&plus;2.7I

188.9091729&plus;104.0046999I

(15)

References

  

Lewin, L. Polylogarithms and Associated Functions. Amsterdam: North Holland, 1981.

See Also

assume

combine/polylog

diff

dilog

evalf

expand

initialfunctions

RealRange

 


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