DLMF:13.23.E5 (Q4639)

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DLMF:13.23.E5
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    ∫ 0 ∞ e 1 2 ⁢ t ⁢ t ν - 1 ⁢ W κ , μ ⁡ ( t ) ⁢ d t = Γ ⁡ ( 1 2 + μ + ν ) ⁢ Γ ⁡ ( 1 2 - μ + ν ) ⁢ Γ ⁡ ( - κ - ν ) Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) , superscript subscript 0 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝜈 1 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑡 𝑡 Euler-Gamma 1 2 𝜇 𝜈 Euler-Gamma 1 2 𝜇 𝜈 Euler-Gamma 𝜅 𝜈 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}W_{% \kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\frac{1}{2}+\mu+\nu% \right)\Gamma\left(\frac{1}{2}-\mu+\nu\right)\Gamma\left(-\kappa-\nu\right)}{% \Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)},}}
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    DLMF:13.23.E5
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