Formula:DLMF:25.12:E5

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formula in Polylogarithms

Formula:DLMF:25.12:E6 >>

${\displaystyle{\displaystyle{\displaystyle\Dilogarithm@{z^{m}}=m\sum_{k=0}^{m-1}\Dilogarithm@{z{\mathrm{e}^{2\pi\mathrm{i}k/m}}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle m=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\{z\in\Complex:|z|<1\}\setminus[0,1)}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\mathrm{Li}_{2}}}}$ : dilogarithm : http://dlmf.nist.gov/25.12#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\in}}}$ : element of : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9
${\displaystyle{\displaystyle{\displaystyle\setminus}}}$ : set subtraction : http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19

Bibliography

Equation (5), Section 25.12 of DLMF.

URL links

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<< Formula:DLMF:25.12:E4

formula in Polylogarithms

Formula:DLMF:25.12:E6 >>