DLMF:25.11.E28 (Q7702): Difference between revisions

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Property / constraint
 

s > - ( 2 n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}}

\realpart@@{s}>-(2n+1)
Property / constraint: s > - ( 2 n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}} / rank
 
Normal rank

Revision as of 18:17, 30 December 2019

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DLMF:25.11.E28
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    ζ ( s , a ) = 1 2 a - s + a 1 - s s - 1 + k = 1 n B 2 k ( 2 k ) ! ( s ) 2 k - 1 a 1 - s - 2 k + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 - k = 1 n B 2 k ( 2 k ) ! x 2 k - 1 ) x s - 1 e - a x d x , Hurwitz-zeta 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript 𝑎 1 𝑠 𝑠 1 superscript subscript 𝑘 1 𝑛 Bernoulli-number-B 2 𝑘 2 𝑘 Pochhammer 𝑠 2 𝑘 1 superscript 𝑎 1 𝑠 2 𝑘 1 Euler-Gamma 𝑠 superscript subscript 0 1 superscript 𝑒 𝑥 1 1 𝑥 1 2 superscript subscript 𝑘 1 𝑛 Bernoulli-number-B 2 𝑘 2 𝑘 superscript 𝑥 2 𝑘 1 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1% -s}}{s-1}+\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+% \frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1% }{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-% ax}\mathrm{d}x,}}
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    DLMF:25.11.E28
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    s > - ( 2 n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}}
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