DLMF:15.4.E13 (Q4997)

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DLMF:15.4.E13
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    Statements

    Defining formula
    F ⁡ ( a , 1 - a ; 1 2 ; - z 2 ) = 1 2 ⁢ 1 + z 2 ⁢ ( ( 1 + z 2 + z ) 2 ⁢ a - 1 + ( 1 + z 2 - z ) 2 ⁢ a - 1 ) , Gauss-hypergeometric-F 𝑎 1 𝑎 1 2 superscript 𝑧 2 1 2 1 superscript 𝑧 2 superscript 1 superscript 𝑧 2 𝑧 2 𝑎 1 superscript 1 superscript 𝑧 2 𝑧 2 𝑎 1 {\displaystyle{\displaystyle F\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{% 2\sqrt{1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}% }-z\right)^{2a-1}\right),}}
    0 references
    DLMF id
    DLMF:15.4.E13
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2aldec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1ldec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1gdec
    0 references
     
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