Formula:DLMF:25.11:E29

<< Formula:DLMF:25.11:E28

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E30 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}({\mathrm{e}^{2\pi x}}-1)}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{arctan}}}}$ : inverse tangent function : http://dlmf.nist.gov/4.23#SS2.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (29), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E28

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E30 >>