DLMF:25.5.E10 (Q7623)

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DLMF:25.5.E10
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    Statements

    Defining formula
    ζ ⁡ ( s ) = 2 s - 1 1 - 2 1 - s ⁢ ∫ 0 ∞ cos ⁡ ( s ⁢ arctan ⁡ x ) ( 1 + x 2 ) s / 2 ⁢ cosh ⁡ ( 1 2 ⁢ π ⁢ x ) ⁢ d x . Riemann-zeta 𝑠 superscript 2 𝑠 1 1 superscript 2 1 𝑠 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 1 2 𝜋 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_% {0}^{\infty}\frac{\cos\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}% \cosh\left(\frac{1}{2}\pi x\right)}\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:25.5.E10
    0 references
    Symbols used
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2aidec
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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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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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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.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
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    Q10816
    Defining formula
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7adec
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    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1idec
    0 references
     
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