DLMF:15.8.E21 (Q5078)

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DLMF:15.8.E21
No description defined

    Statements

    Defining formula
    F ⁡ ( a , b a - b + 1 ; z ) = ( 1 + z ) - 2 ⁢ a ⁢ F ⁡ ( a , a - b + 1 2 2 ⁢ a - 2 ⁢ b + 1 ; 4 ⁢ z ( 1 + z ) 2 ) , Gauss-hypergeometric-F 𝑎 𝑏 𝑎 𝑏 1 𝑧 superscript 1 𝑧 2 𝑎 Gauss-hypergeometric-F 𝑎 𝑎 𝑏 1 2 2 𝑎 2 𝑏 1 4 𝑧 superscript 1 𝑧 2 {\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=\left(1+\sqrt{z}% \right)^{-2a}F\left({a,a-b+\tfrac{1}{2}\atop 2a-2b+1};\frac{4\sqrt{z}}{(1+% \sqrt{z})^{2}}\right),}}
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    DLMF id
    DLMF:15.8.E21
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    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
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    | ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
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    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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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.m2ajdec
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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.m2akdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1akdec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1tdec
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    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1sdec
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    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1qdec
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