Formula:DLMF:25.11:E31

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Failed to parse (unknown function "\EulerGamma"): {\displaystyle {\displaystyle \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{2 \cosh@@{x}} \diff{x} = 4^{-s} \left( \HurwitzZeta@{s}{\tfrac{1}{4} + \tfrac{1}{4} a} - \HurwitzZeta@{s}{\tfrac{3}{4} + \tfrac{1}{4} a} \right) }}

Constraint(s)

Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{s} > 0}} &
Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{a} > -1}}


Proof

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Use

Failed to parse (unknown function "\HurwitzZeta"): {\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-\expe^{-x}} \diff{x} }} .


Symbols List

& : logical and
 : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
 : integral : http://dlmf.nist.gov/1.4#iv
 : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
 : hyperbolic cosine function : http://dlmf.nist.gov/4.28#E2
 : differential : http://dlmf.nist.gov/1.4#iv
 : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
 : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (31), Section 25.11 of DLMF.

URL links

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