Formula:8313

Digital Library of Mathematical Functions ID 25.5.E7

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

Constraint(s)

${\displaystyle{\displaystyle\Re s>-(2n+1)}}$
${\displaystyle{\displaystyle\Re s>-(2n+1)}}$
${\displaystyle{\displaystyle n=1,2,3,\dots}}$

Symbols List

• ${\displaystyle{\displaystyle B_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$: order not exceeding
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}$: Riemann zeta function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle s}}$: complex variable
 

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