Formula:DLMF:25.11:E31

<< Formula:DLMF:25.11:E30

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E32 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}\expe^{-ax}}{2\cosh x}\diffd x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right)}}}$

Constraint(s)

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

Proof

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Use

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

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\mathrm{cosh}}}}$ : hyperbolic cosine function : http://dlmf.nist.gov/4.28#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (31), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E30

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E32 >>