DLMF:10.22.E53 (Q3427)

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DLMF:10.22.E53
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    Statements

    Defining formula
    ∫ 0 ∞ Y 2 ⁢ ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = - π 2 ⁢ p ⁢ exp ⁡ ( - b 2 8 ⁢ p 2 ) ⁢ ( I ν ⁡ ( b 2 8 ⁢ p 2 ) ⁢ tan ⁡ ( ν ⁢ π ) + 1 π ⁢ K ν ⁡ ( b 2 8 ⁢ p 2 ) ⁢ sec ⁡ ( ν ⁢ π ) ) , superscript subscript 0 Bessel-Y-Weber 2 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 2 𝑝 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 𝜈 𝜋 1 𝜋 modified-Bessel-second-kind 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 𝜈 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}Y_{2\nu}\left(bt\right)\exp\left(% -p^{2}t^{2}\right)\mathrm{d}t=-\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p% ^{2}}\right)\left(I_{\nu}\left(\frac{b^{2}}{8p^{2}}\right)\tan\left(\nu\pi% \right)+\frac{1}{\pi}K_{\nu}\left(\frac{b^{2}}{8p^{2}}\right)\sec\left(\nu\pi% \right)\right),}}
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    DLMF id
    DLMF:10.22.E53
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    constraint
    | ℜ ⁡ ν | < 1 2 𝜈 1 2 {\displaystyle{\displaystyle|\Re\nu|<\tfrac{1}{2}}}
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    ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re(p^{2})>0}}
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    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2ahdec
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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.m2audec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aazdec
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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.m3aazdec
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    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2abdec
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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.m2abdec
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