DLMF:15.8.E22 (Q5079)

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DLMF:15.8.E22
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    Statements

    Defining formula
    F ⁡ ( a , b 1 2 ⁢ ( a + b + 1 ) ; z ) = ( 1 - z - 1 - 1 1 - z - 1 + 1 ) a ⁢ F ⁡ ( a , 1 2 ⁢ ( a + b ) a + b ; 4 ⁢ 1 - z - 1 ( 1 - z - 1 + 1 ) 2 ) , Gauss-hypergeometric-F 𝑎 𝑏 1 2 𝑎 𝑏 1 𝑧 superscript 1 superscript 𝑧 1 1 1 superscript 𝑧 1 1 𝑎 Gauss-hypergeometric-F 𝑎 1 2 𝑎 𝑏 𝑎 𝑏 4 1 superscript 𝑧 1 superscript 1 superscript 𝑧 1 1 2 {\displaystyle{\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=% \left(\frac{\sqrt{1-z^{-1}}-1}{\sqrt{1-z^{-1}}+1}\right)^{a}F\left({a,\tfrac{1% }{2}(a+b)\atop a+b};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}% }\right),}}
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    DLMF id
    DLMF:15.8.E22
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    constraint
    | ph ⁡ ( - z ) | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi}}
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    ℜ ⁡ z < 1 2 𝑧 1 2 {\displaystyle{\displaystyle\Re z<\frac{1}{2}}}
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    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.m2akdec
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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.m2aldec
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