DLMF:5.13.E1 (Q2158)

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DLMF:5.13.E1
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( s + a ) ⁢ Γ ⁡ ( b - s ) ⁢ z - s ⁢ d s = Γ ⁡ ( a + b ) ⁢ z a ( 1 + z ) a + b , 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑠 𝑎 Euler-Gamma 𝑏 𝑠 superscript 𝑧 𝑠 𝑠 Euler-Gamma 𝑎 𝑏 superscript 𝑧 𝑎 superscript 1 𝑧 𝑎 𝑏 {\displaystyle{\displaystyle{\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}% \Gamma\left(s+a\right)\Gamma\left(b-s\right)z^{-s}\mathrm{d}s=\frac{\Gamma% \left(a+b\right)z^{a}}{(1+z)^{a+b}}},}}
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    DLMF id
    DLMF:5.13.E1
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    constraint
    ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}
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    - ℜ ⁡ a < c < ℜ ⁡ b 𝑎 𝑐 𝑏 {\displaystyle{\displaystyle-\Re a<c<\Re b}}
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    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
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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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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.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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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
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    Q11282
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C5.S1.XMD11.m1dec
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    Q11273
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1dec
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    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1dec
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