DLMF:25.5.E7 (Q7620): Difference between revisions

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Property / constraint
 

n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle n=1,2,3,\dots}}

n=1,2,3,\dots
Property / constraint: n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle n=1,2,3,\dots}} / rank
 
Normal rank

Revision as of 17:14, 30 December 2019

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DLMF:25.5.E7
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    Statements

    test
    ζ ⁡ ( s ) = 1 2 + 1 s - 1 + ∑ m = 1 n B 2 ⁢ m ( 2 ⁢ m ) ! ⁢ ( s ) 2 ⁢ m - 1 + 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ ( 1 e x - 1 - 1 x + 1 2 - ∑ m = 1 n B 2 ⁢ m ( 2 ⁢ m ) ! ⁢ x 2 ⁢ m - 1 ) ⁢ x s - 1 e x ⁢ d x , Riemann-zeta 𝑠 1 2 1 𝑠 1 superscript subscript 𝑚 1 𝑛 Bernoulli-number-B 2 𝑚 2 𝑚 Pochhammer 𝑠 2 𝑚 1 1 Euler-Gamma 𝑠 superscript subscript 0 1 superscript 𝑒 𝑥 1 1 𝑥 1 2 superscript subscript 𝑚 1 𝑛 Bernoulli-number-B 2 𝑚 2 𝑚 superscript 𝑥 2 𝑚 1 superscript 𝑥 𝑠 1 superscript 𝑒 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum% _{m=1}^{n}\frac{B_{2m}}{(2m)!}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s% \right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_% {m=1}^{n}\frac{B_{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x,}}
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    DLMF id
    DLMF:25.5.E7
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    constraint
    ℜ ⁡ s > - ( 2 ⁢ n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}}
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    ℜ ⁡ s > - ( 2 ⁢ n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}}
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    n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle n=1,2,3,\dots}}
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