DLMF:12.12.E1 (Q4226)

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DLMF:12.12.E1
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    Statements

    Defining formula
    ∫ 0 ∞ e - 1 4 ⁢ t 2 ⁢ t μ - 1 ⁢ U ⁡ ( a , t ) ⁢ d t = π ⁢ 2 - 1 2 ⁢ ( μ + a + 1 2 ) ⁢ Γ ⁡ ( μ ) Γ ⁡ ( 1 2 ⁢ ( μ + a + 3 2 ) ) , superscript subscript 0 superscript 𝑒 1 4 superscript 𝑡 2 superscript 𝑡 𝜇 1 parabolic-U 𝑎 𝑡 𝑡 𝜋 superscript 2 1 2 𝜇 𝑎 1 2 Euler-Gamma 𝜇 Euler-Gamma 1 2 𝜇 𝑎 3 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}U% \left(a,t\right)\mathrm{d}t=\frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})% }\Gamma\left(\mu\right)}{\Gamma\left(\frac{1}{2}(\mu+a+\frac{3}{2})\right)},}}
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    DLMF id
    DLMF:12.12.E1
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    constraint
    ℜ ⁡ μ > 0 𝜇 0 {\displaystyle{\displaystyle\Re\mu>0}}
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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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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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