DLMF:13.16.E10 (Q4561)

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DLMF:13.16.E10
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    Defining formula
    1 Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ M κ , μ ⁡ ( e ± π ⁢ i ⁢ z ) = e 1 2 ⁢ z ± ( 1 2 + μ ) ⁢ π ⁢ i 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 + μ - κ ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t - κ ) ⁢ Γ ⁡ ( 1 2 + μ - t ) Γ ⁡ ( 1 2 + μ + t ) ⁢ z t ⁢ d t , 1 Euler-Gamma 1 2 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑧 superscript 𝑒 plus-or-minus 1 2 𝑧 1 2 𝜇 𝜋 imaginary-unit 2 𝜋 imaginary-unit Euler-Gamma 1 2 𝜇 𝜅 superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑡 𝜅 Euler-Gamma 1 2 𝜇 𝑡 Euler-Gamma 1 2 𝜇 𝑡 superscript 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}% \left(e^{\pm\pi\mathrm{i}}z\right)=\frac{e^{\frac{1}{2}z\pm(\frac{1}{2}+\mu)% \pi\mathrm{i}}}{2\pi\mathrm{i}\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int% _{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(t-\kappa\right)\Gamma% \left(\frac{1}{2}+\mu-t\right)}{\Gamma\left(\frac{1}{2}+\mu+t\right)}z^{t}% \mathrm{d}t,}}
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    DLMF id
    DLMF:13.16.E10
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