DLMF:12.12.E2 (Q4227)

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DLMF:12.12.E2
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    Statements

    Defining formula
    ∫ 0 ∞ e - 3 4 ⁢ t 2 ⁢ t - a - 3 2 ⁢ U ⁡ ( a , t ) ⁢ d t = 2 1 4 + 1 2 ⁢ a ⁢ Γ ⁡ ( - a - 1 2 ) ⁢ cos ⁡ ( ( 1 4 ⁢ a + 1 8 ) ⁢ π ) , superscript subscript 0 superscript 𝑒 3 4 superscript 𝑡 2 superscript 𝑡 𝑎 3 2 parabolic-U 𝑎 𝑡 𝑡 superscript 2 1 4 1 2 𝑎 Euler-Gamma 𝑎 1 2 1 4 𝑎 1 8 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{% 3}{2}}U\left(a,t\right)\mathrm{d}t=2^{\frac{1}{4}+\frac{1}{2}a}\Gamma\left(-a-% \tfrac{1}{2}\right)\cos\left((\tfrac{1}{4}a+\tfrac{1}{8})\pi\right),}}
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    DLMF id
    DLMF:12.12.E2
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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.m2aadec
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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.m2aadec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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