DLMF:13.23.E6 (Q4640)

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DLMF:13.23.E6
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    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e z ⁢ t + 1 2 ⁢ t - 1 ⁢ t κ ⁢ M κ , μ ⁡ ( t - 1 ) ⁢ d t = z - κ - 1 2 Γ ⁡ ( 1 2 + μ - κ ) ⁢ I 2 ⁢ μ ⁡ ( 2 ⁢ z ) , 1 Euler-Gamma 1 2 𝜇 2 𝜋 imaginary-unit superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 1 superscript 𝑡 𝜅 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 superscript 𝑡 1 𝑡 superscript 𝑧 𝜅 1 2 Euler-Gamma 1 2 𝜇 𝜅 modified-Bessel-first-kind 2 𝜇 2 𝑧 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)2\pi\mathrm{i}}% \int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}M_{\kappa,\mu}\left(t^{% -1}\right)\mathrm{d}t=\frac{z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+% \mu-\kappa\right)}I_{2\mu}\left(2\sqrt{z}\right),}}
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    DLMF:13.23.E6
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