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DLMF:15.4.E28 - MaRDI portal

DLMF:15.4.E28 (Q5709)

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DLMF:15.4.E28
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    Statements

    DLMF defining formula
    F ⁡ ( a , b ; 1 2 ⁢ a + 1 2 ⁢ b + 1 2 ; 1 2 ) = π ⁢ Γ ⁡ ( 1 2 ⁢ a + 1 2 ⁢ b + 1 2 ) Γ ⁡ ( 1 2 ⁢ a + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ b + 1 2 ) . Gauss-hypergeometric-F 𝑎 𝑏 1 2 𝑎 1 2 𝑏 1 2 1 2 𝜋 Euler-Gamma 1 2 𝑎 1 2 𝑏 1 2 Euler-Gamma 1 2 𝑎 1 2 Euler-Gamma 1 2 𝑏 1 2 {\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{% 2};\tfrac{1}{2}\right)=\sqrt{\pi}\frac{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b% +\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left% (\tfrac{1}{2}b+\tfrac{1}{2}\right)}.}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    DLMF defining formula
    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.m2azdec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    real or complex parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1udec
    0 references
    real or complex parameter
    DLMF defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1idec
    0 references

    Identifiers

     
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