DLMF:10.22.E49 (Q3423)

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DLMF:10.22.E49
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    Statements

    Defining formula
    ∫ 0 ∞ t μ - 1 ⁢ e - a ⁢ t ⁢ J ν ⁡ ( b ⁢ t ) ⁢ d t = ( 1 2 ⁢ b ) ν a μ + ν ⁢ Γ ⁡ ( μ + ν ) ⁢ 𝐅 ⁡ ( μ + ν 2 , μ + ν + 1 2 ; ν + 1 ; - b 2 a 2 ) , superscript subscript 0 superscript 𝑡 𝜇 1 superscript 𝑒 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 𝑡 superscript 1 2 𝑏 𝜈 superscript 𝑎 𝜇 𝜈 Euler-Gamma 𝜇 𝜈 scaled-hypergeometric-bold-F 𝜇 𝜈 2 𝜇 𝜈 1 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt% \right)\mathrm{d}t=\frac{(\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+% \nu\right)\*\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac% {b^{2}}{a^{2}}\right),}}
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    DLMF id
    DLMF:10.22.E49
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    constraint
    ℜ ⁡ ( μ + ν ) > 0 , ℜ ⁡ ( a ± i ⁢ b ) > 0 formulae-sequence 𝜇 𝜈 0 plus-or-minus 𝑎 𝑖 𝑏 0 {\displaystyle{\displaystyle\Re(\mu+\nu)>0,\Re(a\pm ib)>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.m2aakdec
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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.m2andec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aavdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aavdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aaddec
    0 references
     
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