DLMF:10.43.E30 (Q3644)

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DLMF:10.43.E30
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    Statements

    Defining formula
    f ⁢ ( y ) = 2 ⁢ y π 2 ⁢ sinh ⁡ ( π ⁢ y ) ⁢ ∫ 0 ∞ g ⁢ ( x ) x ⁢ K i ⁢ y ⁡ ( x ) ⁢ d x . 𝑓 𝑦 2 𝑦 superscript 𝜋 2 𝜋 𝑦 superscript subscript 0 𝑔 𝑥 𝑥 modified-Bessel-second-kind 𝑖 𝑦 𝑥 𝑥 {\displaystyle{\displaystyle f(y)=\frac{2y}{\pi^{2}}\sinh\left(\pi y\right)% \int_{0}^{\infty}\frac{g(x)}{x}K_{iy}\left(x\right)\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:10.43.E30
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aydec
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    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aydec
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    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2aodec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1mdec
    0 references
     
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