DLMF:10.22.E67 (Q3441)

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DLMF:10.22.E67
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ t ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ J ν ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ d t = 1 2 ⁢ p 2 ⁢ exp ⁡ ( - a 2 + b 2 4 ⁢ p 2 ) ⁢ I ν ⁡ ( a ⁢ b 2 ⁢ p 2 ) , superscript subscript 0 𝑡 superscript 𝑝 2 superscript 𝑡 2 Bessel-J 𝜈 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 1 2 superscript 𝑝 2 superscript 𝑎 2 superscript 𝑏 2 4 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 𝑎 𝑏 2 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{% \nu}\left(at\right)J_{\nu}\left(bt\right)\mathrm{d}t=\frac{1}{2p^{2}}\exp\left% (-\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{\nu}\left(\frac{ab}{2p^{2}}\right),}}
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    DLMF id
    DLMF:10.22.E67
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    constraint
    ℜ ⁡ ν > - 1 , ℜ ⁡ ( p 2 ) > 0 formulae-sequence 𝜈 1 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re\nu>-1,\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.m2aazdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abndec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2afdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abndec
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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.m2acdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aandec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1atdec
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