DLMF:10.22.E68 (Q3442)

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DLMF:10.22.E68
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    Statements

    Defining formula
    ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ J 0 ⁡ ( a ⁢ t ) ⁢ Y 0 ⁡ ( a ⁢ t ) ⁢ d t = - 1 2 ⁢ π ⁢ p 2 ⁢ exp ⁡ ( - a 2 2 ⁢ p 2 ) ⁢ K 0 ⁡ ( a 2 2 ⁢ p 2 ) , superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 Bessel-J 0 𝑎 𝑡 Bessel-Y-Weber 0 𝑎 𝑡 𝑡 1 2 𝜋 superscript 𝑝 2 superscript 𝑎 2 2 superscript 𝑝 2 modified-Bessel-second-kind 0 superscript 𝑎 2 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{0% }\left(at\right)Y_{0}\left(at\right)\mathrm{d}t=-\frac{1}{2\pi p^{2}}\exp\left% (-\frac{a^{2}}{2p^{2}}\right)K_{0}\left(\frac{a^{2}}{2p^{2}}\right),}}
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    DLMF id
    DLMF:10.22.E68
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    constraint
    ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re(p^{2})>0}}
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2abadec
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    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.m2ajdec
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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.m2azdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abodec
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