DLMF:5.9.E3 (Q2092)

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DLMF:5.9.E3
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    Statements

    Defining formula
    c - z ⁢ Γ ⁡ ( z ) = ∫ - ∞ ∞ | t | 2 ⁢ z - 1 ⁢ e - c ⁢ t 2 ⁢ d t , superscript 𝑐 𝑧 Euler-Gamma 𝑧 superscript subscript superscript 𝑡 2 𝑧 1 superscript 𝑒 𝑐 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle c^{-z}\Gamma\left(z\right)=\int_{-\infty}^{\infty% }|t|^{2z-1}e^{-ct^{2}}\mathrm{d}t,}}
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    DLMF id
    DLMF:5.9.E3
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    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    c > 0 𝑐 0 {\displaystyle{\displaystyle c>0}}
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    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>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.m2abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
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