DLMF:13.23.E10 (Q4644)

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DLMF:13.23.E10
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    Statements

    Defining formula
    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ ∫ 0 ∞ e - 1 2 ⁢ t ⁢ t 1 2 ⁢ ( ν - 1 ) - μ ⁢ M κ , μ ⁡ ( t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = e - 1 2 ⁢ x ⁢ x 1 2 ⁢ ( κ + μ - 3 2 ) Γ ⁡ ( 1 2 + μ + κ ) ⁢ W 1 2 ⁢ ( κ - 3 ⁢ μ + ν + 1 2 ) , 1 2 ⁢ ( κ + μ - ν - 1 2 ) ⁡ ( x ) , 1 Euler-Gamma 1 2 𝜇 superscript subscript 0 superscript 𝑒 1 2 𝑡 superscript 𝑡 1 2 𝜈 1 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 superscript 𝑒 1 2 𝑥 superscript 𝑥 1 2 𝜅 𝜇 3 2 Euler-Gamma 1 2 𝜇 𝜅 Whittaker-confluent-hypergeometric-W 1 2 𝜅 3 𝜇 𝜈 1 2 1 2 𝜅 𝜇 𝜈 1 2 𝑥 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{% \infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}M_{\kappa,\mu}\left(t\right)% J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{e^{-\frac{1}{2}x}x^{\frac{1}{2% }(\kappa+\mu-\frac{3}{2})}}{\Gamma\left(\frac{1}{2}+\mu+\kappa\right)}\*W_{% \frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2}),\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2% })}\left(x\right),}}
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    DLMF id
    DLMF:13.23.E10
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    constraint
    - 1 < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ ( μ + κ ) + 1 2 1 𝜈 2 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(\mu+\kappa)+\tfrac{1}{2}}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    - 1 < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ ( μ + κ ) + 1 2 1 𝜈 2 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(\mu+\kappa)+\tfrac{1}{2}}}
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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.m2aadec
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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.m2aidec
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