DLMF:10.43.E29 (Q3643)

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DLMF:10.43.E29
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ I 0 ⁡ ( a ⁢ t ) ⁢ K 0 ⁡ ( a ⁢ t ) ⁢ d t = 1 4 ⁢ 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 modified-Bessel-first-kind 0 𝑎 𝑡 modified-Bessel-second-kind 0 𝑎 𝑡 𝑡 1 4 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)I_{0% }\left(at\right)K_{0}\left(at\right)\mathrm{d}t=\frac{1}{4p^{2}}\exp\left(% \frac{a^{2}}{2p^{2}}\right)K_{0}\left(\frac{a^{2}}{2p^{2}}\right),}}
    0 references
    DLMF id
    DLMF:10.43.E29
    0 references
    constraint
    ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re\left(p^{2}\right)>0}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1axdec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2addec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3axdec
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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.m2agdec
    0 references
    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.m2andec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aldec
    0 references
     
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