DLMF:8.14.E1 (Q2634)

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DLMF:8.14.E1
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    Statements

    Defining formula
    ∫ 0 ∞ e - a ⁢ x ⁢ γ ⁡ ( b , x ) Γ ⁡ ( b ) ⁢ d x = ( 1 + a ) - b a , superscript subscript 0 superscript 𝑒 𝑎 𝑥 incomplete-gamma 𝑏 𝑥 Euler-Gamma 𝑏 𝑥 superscript 1 𝑎 𝑏 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x% \right)}{\Gamma\left(b\right)}\mathrm{d}x=\frac{(1+a)^{-b}}{a},}}
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    DLMF id
    DLMF:8.14.E1
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    ℜ ⁡ b > - 1 𝑏 1 {\displaystyle{\displaystyle\Re b>-1}}
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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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    incomplete gamma function
    Defining formula
    γ ⁡ ( a , z ) incomplete-gamma 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E1.m2adec
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