DLMF:25.6.E6 (Q7643)

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DLMF:25.6.E6
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    Statements

    Defining formula
    ζ ⁡ ( 2 ⁢ k + 1 ) = ( - 1 ) k + 1 ⁢ ( 2 ⁢ π ) 2 ⁢ k + 1 2 ⁢ ( 2 ⁢ k + 1 ) ! ⁢ ∫ 0 1 B 2 ⁢ k + 1 ⁡ ( t ) ⁢ cot ⁡ ( π ⁢ t ) ⁢ d t , Riemann-zeta 2 𝑘 1 superscript 1 𝑘 1 superscript 2 𝜋 2 𝑘 1 2 2 𝑘 1 superscript subscript 0 1 Bernoulli-polynomial-B 2 𝑘 1 𝑡 𝜋 𝑡 𝑡 {\displaystyle{\displaystyle\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+% 1}}{2(2k+1)!}\int_{0}^{1}B_{2k+1}\left(t\right)\cot\left(\pi t\right)\mathrm{d% }t,}}
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    DLMF id
    DLMF:25.6.E6
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    constraint
    k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle k=1,2,3,\dots}}
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    Symbols used
    Q11288
    Defining formula
    B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS1.m2adec
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    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2aedec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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