DLMF:15.6.E9 (Q5047)

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DLMF:15.6.E9
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    Statements

    test
    𝐅 ⁑ ( a , b ; c ; z ) = ∫ 0 1 t d - 1 ⁒ ( 1 - t ) c - d - 1 ( 1 - z ⁒ t ) a + b - Ξ» ⁒ 𝐅 ⁑ ( Ξ» - a , Ξ» - b d ; z ⁒ t ) ⁒ 𝐅 ⁑ ( a + b - Ξ» , Ξ» - d c - d ; ( 1 - t ) ⁒ z 1 - z ⁒ t ) ⁒ d t , scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript subscript 0 1 superscript 𝑑 𝑑 1 superscript 1 𝑑 𝑐 𝑑 1 superscript 1 𝑧 𝑑 π‘Ž 𝑏 πœ† scaled-hypergeometric-bold-F πœ† π‘Ž πœ† 𝑏 𝑑 𝑧 𝑑 scaled-hypergeometric-bold-F π‘Ž 𝑏 πœ† πœ† 𝑑 𝑐 𝑑 1 𝑑 𝑧 1 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t% ^{d-1}(1-t)^{c-d-1}}{(1-zt)^{a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b% \atop d};zt\right)\mathbf{F}\left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)% z}{1-zt}\right)\mathrm{d}t,}}
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    DLMF id
    DLMF:15.6.E9
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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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    Ξ» ∈ β„‚ πœ† {\displaystyle{\displaystyle\lambda\in\mathbb{C}}}
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    β„œ ⁑ c > β„œ ⁑ d > 0 𝑐 𝑑 0 {\displaystyle{\displaystyle\Re c>\Re d>0}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    Q10923
    test
    β„‚ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
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    Q10770
    test
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
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    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
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    Q10811
    test
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
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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.m2ahdec
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    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1hdec
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    Q11645
    test
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1hdec
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    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1hdec
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    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1hdec
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