DLMF:15.6.E7 (Q5045)

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DLMF:15.6.E7
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    Statements

    test
    𝐅 ⁑ ( a , b ; c ; z ) = 1 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c - a ) ⁒ Ξ“ ⁑ ( c - b ) ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( a + t ) ⁒ Ξ“ ⁑ ( b + t ) ⁒ Ξ“ ⁑ ( c - a - b - t ) ⁒ Ξ“ ⁑ ( - t ) ⁒ ( 1 - z ) t ⁒ d t , scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 1 2 πœ‹ imaginary-unit Euler-Gamma π‘Ž Euler-Gamma 𝑏 Euler-Gamma 𝑐 π‘Ž Euler-Gamma 𝑐 𝑏 superscript subscript imaginary-unit imaginary-unit Euler-Gamma π‘Ž 𝑑 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑐 π‘Ž 𝑏 𝑑 Euler-Gamma 𝑑 superscript 1 𝑧 𝑑 𝑑 {\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 ) | < Ο€ phase 1 𝑧 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    | ph ⁑ ( 1 - z ) | < Ο€ ph 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    a , b , c - a , c - b β‰  0 , - 1 , - 2 , … formulae-sequence π‘Ž 𝑏 𝑐 π‘Ž 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
    test
    Ξ“ ⁑ ( 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
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    Q10770
    test
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1afdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    test
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2afdec
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    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1fdec
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    Q11645
    test
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1fdec
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    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1fdec
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    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1fdec
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