DLMF:24.7.E10 (Q7486)

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DLMF:24.7.E10
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    Statements

    Defining formula
    E 2 ⁢ n + 1 ⁡ ( x ) = ( - 1 ) n + 1 ⁢ 4 ⁢ ∫ 0 ∞ cos ⁡ ( π ⁢ x ) ⁢ sinh ⁡ ( π ⁢ t ) cosh ⁡ ( 2 ⁢ π ⁢ t ) - cos ⁡ ( 2 ⁢ π ⁢ x ) ⁢ t 2 ⁢ n + 1 ⁢ d t . Euler-polynomial-E 2 𝑛 1 𝑥 superscript 1 𝑛 1 4 superscript subscript 0 𝜋 𝑥 𝜋 𝑡 2 𝜋 𝑡 2 𝜋 𝑥 superscript 𝑡 2 𝑛 1 𝑡 {\displaystyle{\displaystyle E_{2n+1}\left(x\right)=(-1)^{n+1}4\*\int_{0}^{% \infty}\frac{\cos\left(\pi x\right)\sinh\left(\pi t\right)}{\cosh\left(2\pi t% \right)-\cos\left(2\pi x\right)}t^{2n+1}\mathrm{d}t.}}
    0 references
    DLMF id
    DLMF:24.7.E10
    0 references
    Symbols used
    Q12114
    Defining formula
    E n ⁡ ( x ) Euler-polynomial-E 𝑛 𝑥 {\displaystyle{\displaystyle E_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS2.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2acdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
    0 references
    Q12110
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C24.S1.XMD5.m1idec
    0 references
    Q12111
    Defining formula
    t 𝑡 {\displaystyle{\displaystyle t}}
    xml-id
    C24.S1.XMD6.m1idec
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