Formula:8402

Digital Library of Mathematical Functions ID 25.11.E35

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "TeX input is invalid."): {\displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\diff{x}=2^{-s}\left(\Hurwitzzeta@{s}{\tfrac{1}{2}a}-\Hurwitzzeta@{s}{\tfrac{1}{2}(1+a)}\right),}

Constraint(s)

${\displaystyle{\displaystyle\Re a>0}}$
${\displaystyle{\displaystyle\Re s>0}}$
${\displaystyle{\displaystyle\Re a=0}}$
${\displaystyle{\displaystyle\Im a\neq 0}}$
${\displaystyle{\displaystyle 0<\Re s<1}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$: Hurwitz zeta function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\Im}}$: imaginary part
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
• ${\displaystyle{\displaystyle s}}$: complex variable
 

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