Formula:DLMF:25.12:E9

<< Formula:DLMF:25.12:E8

formula in Polylogarithms

Formula:DLMF:25.12:E10 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}\ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\mathrm{d}x}}}$

Constraint(s)

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

Note(s)

The right-hand side is called \emph{Clausen's integral}

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4

Bibliography

Equation (9), Section 25.12 of DLMF.

URL links

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