DLMF:13.23.E12 (Q4646)

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DLMF:13.23.E12
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    Statements

    Defining formula
    ∫ 0 ∞ e - 1 2 ⁢ t ⁢ t 1 2 ⁢ ( ν - 1 ) - μ ⁢ W κ , μ ⁡ ( t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = Γ ⁡ ( ν - 2 ⁢ μ + 1 ) Γ ⁡ ( 3 2 - μ - κ + ν ) ⁢ e - 1 2 ⁢ x ⁢ x 1 2 ⁢ ( μ + κ - 3 2 ) ⁢ M 1 2 ⁢ ( κ - 3 ⁢ μ + ν + 1 2 ) , 1 2 ⁢ ( ν - μ - κ + 1 2 ) ⁡ ( x ) , superscript subscript 0 superscript 𝑒 1 2 𝑡 superscript 𝑡 1 2 𝜈 1 𝜇 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 Euler-Gamma 𝜈 2 𝜇 1 Euler-Gamma 3 2 𝜇 𝜅 𝜈 superscript 𝑒 1 2 𝑥 superscript 𝑥 1 2 𝜇 𝜅 3 2 Whittaker-confluent-hypergeometric-M 1 2 𝜅 3 𝜇 𝜈 1 2 1 2 𝜈 𝜇 𝜅 1 2 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(% \nu-1)-\mu}W_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d% }t=\frac{\Gamma\left(\nu-2\mu+1\right)}{\Gamma\left(\frac{3}{2}-\mu-\kappa+\nu% \right)}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*M_{\frac{1% }{2}(\kappa-3\mu+\nu+\frac{1}{2}),\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}% \left(x\right),}}
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    DLMF id
    DLMF:13.23.E12
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    constraint
    max ⁡ ( 2 ⁢ ℜ ⁡ μ - 1 , - 1 ) < ℜ ⁡ ν 2 𝜇 1 1 𝜈 {\displaystyle{\displaystyle\max(2\Re\mu-1,-1)<\Re\nu}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    max ⁡ ( 2 ⁢ ℜ ⁡ μ - 1 , - 1 ) < ℜ ⁡ ν 2 𝜇 1 1 𝜈 {\displaystyle{\displaystyle\max(2\Re\mu-1,-1)<\Re\nu}}
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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.m2acdec
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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.m2akdec
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    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2agdec
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    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
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