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

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Property / Symbols used
 
Property / Symbols used: order not exceeding / rank
 
Normal rank
Property / Symbols used: order not exceeding / qualifier
 
test:

O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}

\bigO@{\NVar{x}}
Property / Symbols used: order not exceeding / qualifier
 
xml-id: C2.S1.E3.m2adec

Revision as of 12:25, 2 January 2020

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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,}}
    0 references
    DLMF id
    DLMF:25.5.E7
    0 references
    constraint
    ℜ ⁡ s > - ( 2 ⁢ n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}}
    0 references
    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}}
    0 references
    Symbols used
    Q11109
    test
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1adec
    0 references
    order not exceeding
    test
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
     
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