DLMF:15.6.E7 (Q5045)

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DLMF:15.6.E7
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    Statements

    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) = 1 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( c - b ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( a + t ) ⁢ Γ ⁡ ( b + t ) ⁢ Γ ⁡ ( c - a - b - t ) ⁢ Γ ⁡ ( - t ) ⁢ ( 1 - z ) t ⁢ d t , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 1 2 𝜋 imaginary-unit Euler-Gamma 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑐 𝑏 superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑎 𝑡 Euler-Gamma 𝑏 𝑡 Euler-Gamma 𝑐 𝑎 𝑏 𝑡 Euler-Gamma 𝑡 superscript 1 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi% \mathrm{i}\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left(c-a\right)\Gamma% \left(c-b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+t% \right)\Gamma\left(b+t\right)\Gamma\left(c-a-b-t\right)\Gamma\left(-t\right)(1% -z)^{t}\mathrm{d}t,}}
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    DLMF id
    DLMF:15.6.E7
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    constraint
    | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    | ph ⁡ ( 1 - z ) | < π ph 1 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    a , b , c - a , c - b ≠ 0 , - 1 , - 2 , … formulae-sequence 𝑎 𝑏 𝑐 𝑎 𝑐 𝑏 0 1 2 … {\displaystyle{\displaystyle a,b,c-a,c-b\neq 0,-1,-2,\dots}}
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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.m2afdec
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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.m2afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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