DLMF:15.4.E30 (Q5014)

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DLMF:15.4.E30
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    Statements

    test
    F ⁡ ( a , 1 - a ; b ; 1 2 ) = 2 1 - b ⁢ π ⁢ Γ ⁡ ( b ) Γ ⁡ ( 1 2 ⁢ a + 1 2 ⁢ b ) ⁢ Γ ⁡ ( 1 2 ⁢ b - 1 2 ⁢ a + 1 2 ) . Gauss-hypergeometric-F 𝑎 1 𝑎 𝑏 1 2 superscript 2 1 𝑏 𝜋 Euler-Gamma 𝑏 Euler-Gamma 1 2 𝑎 1 2 𝑏 Euler-Gamma 1 2 𝑏 1 2 𝑎 1 2 {\displaystyle{\displaystyle F\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{1-b}% \sqrt{\pi}\Gamma\left(b\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)% \Gamma\left(\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}\right)}.}}
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    DLMF id
    DLMF:15.4.E30
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ahdec
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    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    test
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2aabdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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