# Formula:DLMF:25.11:E31

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \realpart{s} > 0}}**&

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \realpart{a} > -1}}**

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Gamma}}**
: Euler's gamma function : http://dlmf.nist.gov/5.2#E1

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \int}}**
: integral : http://dlmf.nist.gov/1.4#iv

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \mathrm{e}}}**
: the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \mathrm{cosh}}}**
: hyperbolic cosine function : http://dlmf.nist.gov/4.28#E2

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \mathrm{d}^nx}}**
: differential : http://dlmf.nist.gov/1.4#iv

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \zeta}}**
: Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\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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