DLMF:13.23.E4 (Q4638)

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DLMF:13.23.E4
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    Statements

    Defining formula
    ∫ 0 ∞ e - z ⁢ t ⁢ t ν - 1 ⁢ W κ , μ ⁡ ( t ) ⁢ d t = Γ ⁡ ( 1 2 + μ + ν ) ⁢ Γ ⁡ ( 1 2 - μ + ν ) ⁢ 𝐅 1 2 ⁡ ( 1 2 - μ + ν , 1 2 + μ + ν ν - κ + 1 ; 1 2 - z ) , superscript subscript 0 superscript 𝑒 𝑧 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 hypergeometric-bold-pFq 2 1 1 2 𝜇 𝜈 1 2 𝜇 𝜈 𝜈 𝜅 1 1 2 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-zt}t^{\nu-1}W_{\kappa,\mu}% \left(t\right)\mathrm{d}t=\Gamma\left(\tfrac{1}{2}+\mu+\nu\right)\Gamma\left(% \tfrac{1}{2}-\mu+\nu\right)\*{{}_{2}{\mathbf{F}}_{1}}\left({\tfrac{1}{2}-\mu+% \nu,\tfrac{1}{2}+\mu+\nu\atop\nu-\kappa+1};\tfrac{1}{2}-z\right),}}
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    DLMF id
    DLMF:13.23.E4
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    constraint
    ℜ ⁡ ( ν + 1 2 ) > | ℜ ⁡ μ | 𝜈 1 2 𝜇 {\displaystyle{\displaystyle\Re(\nu+\tfrac{1}{2})>|\Re\mu|}}
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