DLMF:2.6.E4 (Q829)

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DLMF:2.6.E4
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    Statements

    Defining formula
    ∫ 0 ∞ t α - 1 ( x + t ) α + β ⁢ d t = Γ ⁡ ( α ) ⁢ Γ ⁡ ( β ) Γ ⁡ ( α + β ) ⁢ 1 x β , superscript subscript 0 superscript 𝑡 𝛼 1 superscript 𝑥 𝑡 𝛼 𝛽 𝑡 Euler-Gamma 𝛼 Euler-Gamma 𝛽 Euler-Gamma 𝛼 𝛽 1 superscript 𝑥 𝛽 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha% +\beta}}\mathrm{d}t=\frac{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}{% \Gamma\left(\alpha+\beta\right)}\frac{1}{x^{\beta}},}}
    0 references
    DLMF id
    DLMF:2.6.E4
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    constraint
    ℜ ⁡ α > 0 𝛼 0 {\displaystyle{\displaystyle\Re\alpha>0}}
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    ℜ ⁡ β > 0 𝛽 0 {\displaystyle{\displaystyle\Re\beta>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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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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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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