DLMF:10.43.E25 (Q3639)

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DLMF:10.43.E25
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ K ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = π 4 ⁢ p ⁢ sec ⁡ ( 1 2 ⁢ π ⁢ ν ) ⁢ exp ⁡ ( b 2 8 ⁢ p 2 ) ⁢ K 1 2 ⁢ ν ⁡ ( b 2 8 ⁢ p 2 ) , superscript subscript 0 modified-Bessel-second-kind 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 4 𝑝 1 2 𝜋 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-second-kind 1 2 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}K_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{4p}\sec\left(\tfrac{1}{2}\pi\nu% \right)\exp\left(\frac{b^{2}}{8p^{2}}\right)K_{\frac{1}{2}\nu}\left(\frac{b^{2% }}{8p^{2}}\right),}}
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    DLMF id
    DLMF:10.43.E25
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    constraint
    | ℜ ⁡ ν | < 1 𝜈 1 {\displaystyle{\displaystyle|\Re\nu|<1}}
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    ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re(p^{2})>0}}
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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.m2agdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1atdec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3atdec
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