DLMF:15.8.E2 (Q5059)

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DLMF:15.8.E2
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    Statements

    Defining formula
    sin ⁡ ( π ⁢ ( b - a ) ) π ⁢ 𝐅 ⁡ ( a , b c ; z ) = ( - z ) - a Γ ⁡ ( b ) ⁢ Γ ⁡ ( c - a ) ⁢ 𝐅 ⁡ ( a , a - c + 1 a - b + 1 ; 1 z ) - ( - z ) - b Γ ⁡ ( a ) ⁢ Γ ⁡ ( c - b ) ⁢ 𝐅 ⁡ ( b , b - c + 1 b - a + 1 ; 1 z ) , 𝜋 𝑏 𝑎 𝜋 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 superscript 𝑧 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑐 𝑎 scaled-hypergeometric-bold-F 𝑎 𝑎 𝑐 1 𝑎 𝑏 1 1 𝑧 superscript 𝑧 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑐 𝑏 scaled-hypergeometric-bold-F 𝑏 𝑏 𝑐 1 𝑏 𝑎 1 1 𝑧 {\displaystyle{\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}% \left({a,b\atop c};z\right)=\frac{(-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c% -a\right)}\mathbf{F}\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)-\frac{(-z)^{% -b}}{\Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,b-c+1\atop b% -a+1};\frac{1}{z}\right),}}
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    DLMF id
    DLMF:15.8.E2
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    constraint
    | ph ⁡ ( - z ) | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-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.m2adec
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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.m2aadec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
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