DLMF:12.5.E3 (Q4117)

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DLMF:12.5.E3
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    Statements

    Defining formula
    U ⁡ ( a , z ) = e - 1 4 ⁢ z 2 Γ ⁡ ( 3 4 + 1 2 ⁢ a ) ⁢ ∫ 0 ∞ t 1 2 ⁢ a - 1 4 ⁢ e - t ⁢ ( z 2 + 2 ⁢ t ) - 1 2 ⁢ a - 1 4 ⁢ d t , parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 3 4 1 2 𝑎 superscript subscript 0 superscript 𝑡 1 2 𝑎 1 4 superscript 𝑒 𝑡 superscript superscript 𝑧 2 2 𝑡 1 2 𝑎 1 4 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{% \Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)}\*\int_{0}^{\infty}t^{\frac{1}{2}a% -\frac{1}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{1}{4}}\mathrm{d}% t,}}
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    DLMF id
    DLMF:12.5.E3
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    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
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