DLMF:15.4.E21 (Q5005): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11646 / rank
 
Normal rank
Property / Symbols used: Q11646 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11646 / qualifier
 
xml-id: C15.S1.XMD5.m1cdec

Latest revision as of 13:49, 2 January 2020

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DLMF:15.4.E21
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    Statements

    Defining formula
    lim z → 1 - ⁡ F ⁡ ( a , b ; a + b ; z ) - ln ⁡ ( 1 - z ) = Γ ⁡ ( a + b ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) . subscript → 𝑧 limit-from 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑎 𝑏 𝑧 1 𝑧 Euler-Gamma 𝑎 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑏 {\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;a+b;z\right)}{-\ln% \left(1-z\right)}=\frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma% \left(b\right)}.}}
    0 references
    DLMF id
    DLMF:15.4.E21
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2atdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1sdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1odec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1cdec
    0 references
     
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