DLMF:5.14.E1 (Q2163)

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DLMF:5.14.E1
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    Statements

    Defining formula
    ∫ V n t 1 z 1 - 1 ⁢ t 2 z 2 - 1 ⁢ ⋯ ⁢ t n z n - 1 ⁢ d t 1 ⁢ d t 2 ⁢ ⋯ ⁢ d t n = Γ ⁡ ( z 1 ) ⁢ Γ ⁡ ( z 2 ) ⁢ ⋯ ⁢ Γ ⁡ ( z n ) Γ ⁡ ( 1 + z 1 + z 2 + … + z n ) , subscript subscript 𝑉 𝑛 superscript subscript 𝑡 1 subscript 𝑧 1 1 superscript subscript 𝑡 2 subscript 𝑧 2 1 ⋯ superscript subscript 𝑡 𝑛 subscript 𝑧 𝑛 1 subscript 𝑡 1 subscript 𝑡 2 ⋯ subscript 𝑡 𝑛 Euler-Gamma subscript 𝑧 1 Euler-Gamma subscript 𝑧 2 ⋯ Euler-Gamma subscript 𝑧 𝑛 Euler-Gamma 1 subscript 𝑧 1 subscript 𝑧 2 … subscript 𝑧 𝑛 {\displaystyle{\displaystyle\int_{V_{n}}t_{1}^{z_{1}-1}t_{2}^{z_{2}-1}\cdots t% _{n}^{z_{n}-1}\mathrm{d}t_{1}\mathrm{d}t_{2}\cdots\mathrm{d}t_{n}=\frac{\Gamma% \left(z_{1}\right)\Gamma\left(z_{2}\right)\cdots\Gamma\left(z_{n}\right)}{% \Gamma\left(1+z_{1}+z_{2}+\dots+z_{n}\right)},}}
    0 references
    DLMF id
    DLMF:5.14.E1
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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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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q11274
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1dec
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