DLMF:12.5.E1 (Q4115)

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DLMF:12.5.E1
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    Statements

    Defining formula
    U ⁡ ( a , z ) = e - 1 4 ⁢ z 2 Γ ⁡ ( 1 2 + a ) ⁢ ∫ 0 ∞ t a - 1 2 ⁢ e - 1 2 ⁢ t 2 - z ⁢ t ⁢ d t , parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 1 2 𝑎 superscript subscript 0 superscript 𝑡 𝑎 1 2 superscript 𝑒 1 2 superscript 𝑡 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{% \Gamma\left(\frac{1}{2}+a\right)}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1% }{2}t^{2}-zt}\mathrm{d}t,}}
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    DLMF id
    DLMF:12.5.E1
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    constraint
    ℜ ⁡ a > - 1 2 𝑎 1 2 {\displaystyle{\displaystyle\Re a>-\tfrac{1}{2}}}
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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.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
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    Q11513
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1dec
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    Q11516
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C12.S1.XMD6.m1dec
    0 references
     
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