DLMF:15.6.E3 (Q5041)

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DLMF:15.6.E3
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    Statements

    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) = e - b ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 - b ) 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( c - b ) ⁢ ∫ ∞ ( 0 + ) t b - 1 ⁢ ( t + 1 ) a - c ( t - z ⁢ t + 1 ) a ⁢ d t , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 superscript 𝑒 𝑏 𝜋 imaginary-unit Euler-Gamma 1 𝑏 2 𝜋 imaginary-unit Euler-Gamma 𝑐 𝑏 superscript subscript limit-from 0 superscript 𝑡 𝑏 1 superscript 𝑡 1 𝑎 𝑐 superscript 𝑡 𝑧 𝑡 1 𝑎 𝑡 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}% \frac{\Gamma\left(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{% \infty}^{(0+)}\frac{t^{b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\mathrm{d}t,}}
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    DLMF id
    DLMF:15.6.E3
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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 ⁡ ( c - b ) > 0 𝑐 𝑏 0 {\displaystyle{\displaystyle\Re(c-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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    b ≠ 1 , 2 , 3 , … 𝑏 1 2 3 … {\displaystyle{\displaystyle b\neq 1,2,3,\dots}}
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    ℜ ⁡ ( c - b ) > 0 𝑐 𝑏 0 {\displaystyle{\displaystyle\Re(c-b)>0}}
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