DLMF:12.5.E5 (Q4119)

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DLMF:12.5.E5
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    Statements

    Defining formula
    U ⁡ ( a , z ) = Γ ⁡ ( 1 2 - a ) 2 ⁢ π ⁢ i ⁢ e - 1 4 ⁢ z 2 ⁢ ∫ - ∞ ( 0 + ) e z ⁢ t - 1 2 ⁢ t 2 ⁢ t a - 1 2 ⁢ d t , parabolic-U 𝑎 𝑧 Euler-Gamma 1 2 𝑎 2 𝜋 𝑖 superscript 𝑒 1 4 superscript 𝑧 2 superscript subscript limit-from 0 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 2 superscript 𝑡 𝑎 1 2 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{\Gamma\left(\frac{1}{2}-a% \right)}{2\pi i}e^{-\frac{1}{4}z^{2}}\int_{-\infty}^{(0+)}e^{zt-\frac{1}{2}t^{% 2}}t^{a-\frac{1}{2}}\mathrm{d}t,}}
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    DLMF id
    DLMF:12.5.E5
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    constraint
    - π < ph ⁡ t < π 𝜋 phase 𝑡 𝜋 {\displaystyle{\displaystyle-\pi<\operatorname{ph}t<\pi}}
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    a ≠ 1 2 , 3 2 , 5 2 , … 𝑎 1 2 3 2 5 2 … {\displaystyle{\displaystyle a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots}}
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    - π < ph ⁡ t < π 𝜋 ph 𝑡 𝜋 {\displaystyle{\displaystyle-\pi<\operatorname{ph}t<\pi}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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