DLMF:15.6.E2 (Q5040)

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DLMF:15.6.E2
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    Statements

    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) = Γ ⁡ ( 1 + b - c ) 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( b ) ⁢ ∫ 0 ( 1 + ) t b - 1 ⁢ ( t - 1 ) c - b - 1 ( 1 - z ⁢ t ) a ⁢ d t , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 Euler-Gamma 1 𝑏 𝑐 2 𝜋 imaginary-unit Euler-Gamma 𝑏 superscript subscript 0 limit-from 1 superscript 𝑡 𝑏 1 superscript 𝑡 1 𝑐 𝑏 1 superscript 1 𝑧 𝑡 𝑎 𝑡 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1% +b-c\right)}{2\pi\mathrm{i}\Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t% -1)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,}}
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    DLMF id
    DLMF:15.6.E2
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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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    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}}
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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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    c - b ≠ 1 , 2 , 3 , … 𝑐 𝑏 1 2 3 … {\displaystyle{\displaystyle c-b\neq 1,2,3,\dots}}
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    ℜ ⁡ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>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.m2aadec
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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.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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