DLMF:8.14.E5 (Q2638)

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DLMF:8.14.E5
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    Statements

    test
    ∫ 0 ∞ x a - 1 ⁢ e - s ⁢ x ⁢ γ ⁡ ( b , x ) ⁢ d x = Γ ⁡ ( a + b ) b ⁢ ( 1 + s ) a + b ⁢ F ⁡ ( 1 , a + b ; 1 + b ; 1 / ( 1 + s ) ) , superscript subscript 0 superscript 𝑥 𝑎 1 superscript 𝑒 𝑠 𝑥 incomplete-gamma 𝑏 𝑥 𝑥 Euler-Gamma 𝑎 𝑏 𝑏 superscript 1 𝑠 𝑎 𝑏 Gauss-hypergeometric-F 1 𝑎 𝑏 1 𝑏 1 1 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\gamma\left(b,x% \right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{b(1+s)^{a+b}}\*F\left(1,a+b;1% +b;1/(1+s)\right),}}
    0 references
    DLMF id
    DLMF:8.14.E5
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    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
     
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