DLMF:5.14.E2 (Q2164)

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DLMF:5.14.E2
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    Statements

    Defining formula
    ∫ V n ( 1 - ∑ k = 1 n t k ) z n + 1 - 1 ⁢ ∏ k = 1 n t k z k - 1 ⁢ d t k = Γ ⁡ ( z 1 ) ⁢ Γ ⁡ ( z 2 ) ⁢ ⋯ ⁢ Γ ⁡ ( z n + 1 ) Γ ⁡ ( z 1 + z 2 + … + z n + 1 ) . subscript subscript 𝑉 𝑛 superscript 1 superscript subscript 𝑘 1 𝑛 subscript 𝑡 𝑘 subscript 𝑧 𝑛 1 1 superscript subscript product 𝑘 1 𝑛 superscript subscript 𝑡 𝑘 subscript 𝑧 𝑘 1 subscript 𝑡 𝑘 Euler-Gamma subscript 𝑧 1 Euler-Gamma subscript 𝑧 2 ⋯ Euler-Gamma subscript 𝑧 𝑛 1 Euler-Gamma subscript 𝑧 1 subscript 𝑧 2 … subscript 𝑧 𝑛 1 {\displaystyle{\displaystyle\int_{V_{n}}\left(1-\sum_{k=1}^{n}t_{k}\right)^{z_% {n+1}-1}\prod_{k=1}^{n}t_{k}^{z_{k}-1}\mathrm{d}t_{k}=\frac{\Gamma\left(z_{1}% \right)\Gamma\left(z_{2}\right)\cdots\Gamma\left(z_{n+1}\right)}{\Gamma\left(z% _{1}+z_{2}+\dots+z_{n+1}\right)}.}}
    0 references
    DLMF id
    DLMF:5.14.E2
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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.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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    Q11274
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1adec
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1dec
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