Formula:DLMF:25.12:E7

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formula in Polylogarithms

Formula:DLMF:25.12:E8 >>

${\displaystyle{\displaystyle{\displaystyle\Dilogarithm@{{\mathrm{e}^{\mathrm{i}\theta}}}=\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}+\mathrm{i}\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle 0\leq\theta\leq 2\pi}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{Li}_{2}}}}$ : dilogarithm : http://dlmf.nist.gov/25.12#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4

Bibliography

Equation (7), Section 25.12 of DLMF.

URL links

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