DLMF:24.7.E1 (Q7477)

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DLMF:24.7.E1
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    Statements

    Defining formula
    B 2 ⁢ n = ( - 1 ) n + 1 ⁢ 4 ⁢ n 1 - 2 1 - 2 ⁢ n ⁢ ∫ 0 ∞ t 2 ⁢ n - 1 e 2 ⁢ π ⁢ t + 1 ⁢ d t = ( - 1 ) n + 1 ⁢ 2 ⁢ n 1 - 2 1 - 2 ⁢ n ⁢ ∫ 0 ∞ t 2 ⁢ n - 1 ⁢ e - π ⁢ t ⁢ sech ⁡ ( π ⁢ t ) ⁢ d t , Bernoulli-number-B 2 𝑛 superscript 1 𝑛 1 4 𝑛 1 superscript 2 1 2 𝑛 superscript subscript 0 superscript 𝑡 2 𝑛 1 superscript 𝑒 2 𝜋 𝑡 1 𝑡 superscript 1 𝑛 1 2 𝑛 1 superscript 2 1 2 𝑛 superscript subscript 0 superscript 𝑡 2 𝑛 1 superscript 𝑒 𝜋 𝑡 𝜋 𝑡 𝑡 {\displaystyle{\displaystyle B_{2n}=(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{% \infty}\frac{t^{2n-1}}{e^{2\pi t}+1}\mathrm{d}t=(-1)^{n+1}\frac{2n}{1-2^{1-2n}% }\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\operatorname{sech}\left(\pi t\right)% \mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:24.7.E1
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1adec
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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.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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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    hyperbolic secant function
    Defining formula
    sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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