DLMF:15.6.E7 (Q5045): Difference between revisions

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Property / constraint
 

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}}

a,b,c-a,c-b\neq 0,-1,-2,\dots
Property / constraint: 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}} / rank
 
Normal rank

Revision as of 16:55, 30 December 2019

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DLMF:15.6.E7
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    Statements

    test
    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,}}
    0 references
    DLMF id
    DLMF:15.6.E7
    0 references
    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}}
    0 references
     
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