DLMF:10.22.E54 (Q3428)

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DLMF:10.22.E54
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    Statements

    Defining formula
    ∫ 0 ∞ J ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ t μ - 1 ⁢ d t = ( 1 2 ⁢ b / p ) ν ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ ) 2 ⁢ p μ ⁢ exp ⁡ ( - b 2 4 ⁢ p 2 ) ⁢ 𝐌 ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 , ν + 1 , b 2 4 ⁢ p 2 ) , superscript subscript 0 Bessel-J 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 superscript 𝑡 𝜇 1 𝑡 superscript 1 2 𝑏 𝑝 𝜈 Euler-Gamma 1 2 𝜈 1 2 𝜇 2 superscript 𝑝 𝜇 superscript 𝑏 2 4 superscript 𝑝 2 Kummer-confluent-hypergeometric-bold-M 1 2 𝜈 1 2 𝜇 1 𝜈 1 superscript 𝑏 2 4 superscript 𝑝 2 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-% p^{2}t^{2}\right)t^{\mu-1}\mathrm{d}t=\frac{(\tfrac{1}{2}b/p)^{\nu}\Gamma\left% (\tfrac{1}{2}\nu+\tfrac{1}{2}\mu\right)}{2p^{\mu}}\exp\left(-\frac{b^{2}}{4p^{% 2}}\right)\*{\mathbf{M}}\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^% {2}}{4p^{2}}\right),}}
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    DLMF id
    DLMF:10.22.E54
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    constraint
    ℜ ⁡ ( μ + ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu+\nu)>0}}
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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 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.m2aandec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2apdec
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