DLMF:15.8.E28 (Q5085)

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DLMF:15.8.E28
No description defined

    Statements

    Defining formula
    2 ⁢ z ⁢ Γ ⁡ ( - 1 2 ) ⁢ Γ ⁡ ( a + b - 1 2 ) Γ ⁡ ( a - 1 2 ) ⁢ Γ ⁡ ( b - 1 2 ) ⁢ F ⁡ ( a , b ; 3 2 ; z ) = F ⁡ ( 2 ⁢ a - 1 , 2 ⁢ b - 1 ; a + b - 1 2 ; 1 2 - 1 2 ⁢ z ) - F ⁡ ( 2 ⁢ a - 1 , 2 ⁢ b - 1 ; a + b - 1 2 ; 1 2 + 1 2 ⁢ z ) , 2 𝑧 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 3 2 𝑧 Gauss-hypergeometric-F 2 𝑎 1 2 𝑏 1 𝑎 𝑏 1 2 1 2 1 2 𝑧 Gauss-hypergeometric-F 2 𝑎 1 2 𝑏 1 𝑎 𝑏 1 2 1 2 1 2 𝑧 {\displaystyle{\displaystyle\frac{2\sqrt{z}\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{3}{2};z\right)=F\left(2a-1,2b-1;% a+b-\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}\right)-F\left(2a-1,2b-1;a+b% -\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}\right),}}
    0 references
    DLMF id
    DLMF:15.8.E28
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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.m2aldec
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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.m2aqdec
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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.m2ardec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ardec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1aadec
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    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1zdec
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    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1vdec
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