DLMF:15.8.E27 (Q5084)

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DLMF:15.8.E27
No description defined

    Statements

    Defining formula
    2 ⁢ Γ ⁡ ( 1 2 ) ⁢ Γ ⁡ ( a + b + 1 2 ) Γ ⁡ ( a + 1 2 ) ⁢ Γ ⁡ ( b + 1 2 ) ⁢ F ⁡ ( a , b ; 1 2 ; z ) = F ⁡ ( 2 ⁢ a , 2 ⁢ b ; a + b + 1 2 ; 1 2 - 1 2 ⁢ z ) + F ⁡ ( 2 ⁢ a , 2 ⁢ b ; a + b + 1 2 ; 1 2 + 1 2 ⁢ z ) , 2 Euler-Gamma 1 2 Euler-Gamma 𝑎 𝑏 1 2 Euler-Gamma 𝑎 1 2 Euler-Gamma 𝑏 1 2 Gauss-hypergeometric-F 𝑎 𝑏 1 2 𝑧 Gauss-hypergeometric-F 2 𝑎 2 𝑏 𝑎 𝑏 1 2 1 2 1 2 𝑧 Gauss-hypergeometric-F 2 𝑎 2 𝑏 𝑎 𝑏 1 2 1 2 1 2 𝑧 {\displaystyle{\displaystyle\frac{2\Gamma\left(\tfrac{1}{2}\right)\Gamma\left(% a+b+\tfrac{1}{2}\right)}{\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac% {1}{2}\right)}F\left(a,b;\tfrac{1}{2};z\right)=F\left(2a,2b;a+b+\tfrac{1}{2};% \tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}\right)+F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{% 1}{2}+\tfrac{1}{2}\sqrt{z}\right),}}
    0 references
    DLMF id
    DLMF:15.8.E27
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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.m2akdec
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    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    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.m2apdec
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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.m2aqdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aqdec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1zdec
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    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1ydec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1udec
    0 references
     
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