DLMF:12.5.E8 (Q4122)

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DLMF:12.5.E8
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    Defining formula
    U ⁡ ( a , z ) = e - 1 4 ⁢ z 2 ⁢ z - a - 1 2 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 + a ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 + a - 2 ⁢ t ) ⁢ 2 t ⁢ z 2 ⁢ t ⁢ d t , parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 2 𝜋 𝑖 Euler-Gamma 1 2 𝑎 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑎 2 𝑡 superscript 2 𝑡 superscript 𝑧 2 𝑡 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}z^{-% a-\frac{1}{2}}}{2\pi i\Gamma\left(\frac{1}{2}+a\right)}\*\int_{-i\infty}^{i% \infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}+a-2t\right)2^{t}z^{2t}% \mathrm{d}t,}}
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    DLMF id
    DLMF:12.5.E8
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