DLMF:13.23.E11 (Q4645)

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DLMF:13.23.E11
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    ∫ 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 ⁡ ( 1 2 + μ - κ ) ⁢ e 1 2 ⁢ x ⁢ x 1 2 ⁢ ( μ - κ - 3 2 ) ⁢ W 1 2 ⁢ ( κ + 3 ⁢ μ - ν - 1 2 ) , 1 2 ⁢ ( κ - μ + ν + 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 1 2 𝜇 𝜅 superscript 𝑒 1 2 𝑥 superscript 𝑥 1 2 𝜇 𝜅 3 2 Whittaker-confluent-hypergeometric-W 1 2 𝜅 3 𝜇 𝜈 1 2 1 2 𝜅 𝜇 𝜈 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{1}{2}+\mu-\kappa% \right)}\*e^{\frac{1}{2}x}x^{\frac{1}{2}(\mu-\kappa-\frac{3}{2})}\*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:13.23.E11
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