DLMF:15.8.E4 (Q5061)

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DLMF:15.8.E4
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    Statements

    Defining formula
    sin ⁡ ( π ⁢ ( c - a - b ) ) π ⁢ 𝐅 ⁡ ( a , b c ; z ) = 1 Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( c - b ) ⁢ 𝐅 ⁡ ( a , b a + b - c + 1 ; 1 - z ) - ( 1 - z ) c - a - b Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ 𝐅 ⁡ ( c - a , c - b c - a - b + 1 ; 1 - z ) , 𝜋 𝑐 𝑎 𝑏 𝜋 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 1 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑐 𝑏 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑎 𝑏 𝑐 1 1 𝑧 superscript 1 𝑧 𝑐 𝑎 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑏 scaled-hypergeometric-bold-F 𝑐 𝑎 𝑐 𝑏 𝑐 𝑎 𝑏 1 1 𝑧 {\displaystyle{\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}% \left({a,b\atop c};z\right)=\frac{1}{\Gamma\left(c-a\right)\Gamma\left(c-b% \right)}\mathbf{F}\left({a,b\atop a+b-c+1};1-z\right)-\frac{(1-z)^{c-a-b}}{% \Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,c-b\atop c-a-b+1% };1-z\right),}}
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    DLMF id
    DLMF:15.8.E4
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    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
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    | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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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.m2abdec
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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.m2acdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2acdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1cdec
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    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1cdec
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    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1cdec
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