Formula:DLMF:25.11:E27

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E28 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{{\mathrm{e}^{x}}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}^{ax}}}\mathrm{d}x}}}$

Constraint(s)

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

Proof

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Argue as in

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{{\mathrm{e}^{x}}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}^{x}}}\mathrm{d}x}}}$.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (27), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E26

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E28 >>