DLMF:15.4.E23 (Q5007): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
test:

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)}}

\hyperF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
xml-id: C15.S2.E1.m2avdec
Property / Symbols used
 
Property / Symbols used: Q11644 / rank
 
Normal rank
Property / Symbols used: Q11644 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11644 / qualifier
 
xml-id: C15.S1.XMD3.m1udec
Property / Symbols used
 
Property / Symbols used: Q11645 / rank
 
Normal rank
Property / Symbols used: Q11645 / qualifier
 
test:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11645 / qualifier
 
xml-id: C15.S1.XMD4.m1qdec
Property / Symbols used
 
Property / Symbols used: Q11646 / rank
 
Normal rank
Property / Symbols used: Q11646 / qualifier
 
test:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11646 / qualifier
 
xml-id: C15.S1.XMD5.m1edec
Property / Symbols used
 
Property / Symbols used: Q11647 / rank
 
Normal rank
Property / Symbols used: Q11647 / qualifier
 
test:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q11647 / qualifier
 
xml-id: C15.S1.XMD6.m1bdec

Latest revision as of 12:50, 2 January 2020

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DLMF:15.4.E23
No description defined

    Statements

    test
    lim z → 1 - ⁡ F ⁡ ( a , b ; c ; z ) ( 1 - z ) c - a - b = Γ ⁡ ( c ) ⁢ Γ ⁡ ( a + b - c ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) . subscript → 𝑧 limit-from 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript 1 𝑧 𝑐 𝑎 𝑏 Euler-Gamma 𝑐 Euler-Gamma 𝑎 𝑏 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑏 {\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;c;z\right)}{(1-z)^{% c-a-b}}=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a% \right)\Gamma\left(b\right)}.}}
    0 references
    DLMF id
    DLMF:15.4.E23
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    test
    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.m2avdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1udec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1qdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1edec
    0 references
    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1bdec
    0 references
     
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