Formula:DLMF:25.12:E4

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formula in Polylogarithms

Formula:DLMF:25.12:E5 >>

${\displaystyle{\displaystyle{\displaystyle\Dilogarithm@{z}+\Dilogarithm@{\frac{1}{z}}=-\frac{1}{6}{\pi^{2}}-\frac{1}{2}(\ln\left(-z\right))^{2}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle z\in\Complex\setminus[0,\infty)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{Li}_{2}}}}$ : dilogarithm : http://dlmf.nist.gov/25.12#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\in}}}$ : element of : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9
${\displaystyle{\displaystyle{\displaystyle\mathbb{C}}}}$ : set of complex numbers : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1
${\displaystyle{\displaystyle{\displaystyle\setminus}}}$ : set subtraction : http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19

Bibliography

Equation (4), Section 25.12 of DLMF.

URL links

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