DLMF:15.8.E10 (Q5067): Difference between revisions

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Property / constraint
 

| z - 1 | < 1 , | ph ⁡ ( 1 - z ) | < π formulae-sequence 𝑧 1 1 phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi}}

|z-1|<1,|\phase@{1-z}|<\pi
Property / constraint: | z - 1 | < 1 , | ph ⁡ ( 1 - z ) | < π formulae-sequence 𝑧 1 1 phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2afdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2agdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5abdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1agdec
Property / Symbols used
 
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
test:

𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}

\hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
xml-id: C15.S2.E2.m2agdec
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.m1idec
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.m1gdec
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.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11658 / rank
 
Normal rank
Property / Symbols used: Q11658 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11658 / qualifier
 
xml-id: C15.S1.XMD7.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11651 / rank
 
Normal rank
Property / Symbols used: Q11651 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11651 / qualifier
 
xml-id: C15.S1.XMD9.m1ddec

Latest revision as of 13:03, 2 January 2020

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English
DLMF:15.8.E10
No description defined

    Statements

    test
    𝐅 ⁡ ( a , b a + b + m ; z ) = 1 Γ ⁡ ( a + m ) ⁢ Γ ⁡ ( b + m ) ⁢ ∑ k = 0 m - 1 ( a ) k ⁢ ( b ) k ⁢ ( m - k - 1 ) ! k ! ⁢ ( z - 1 ) k - ( z - 1 ) m Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ ∑ k = 0 ∞ ( a + m ) k ⁢ ( b + m ) k k ! ⁢ ( k + m ) ! ⁢ ( 1 - z ) k ⁢ ( ln ⁡ ( 1 - z ) - ψ ⁡ ( k + 1 ) - ψ ⁡ ( k + m + 1 ) + ψ ⁡ ( a + k + m ) + ψ ⁡ ( b + k + m ) ) , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑎 𝑏 𝑚 𝑧 1 Euler-Gamma 𝑎 𝑚 Euler-Gamma 𝑏 𝑚 superscript subscript 𝑘 0 𝑚 1 subscript 𝑎 𝑘 subscript 𝑏 𝑘 𝑚 𝑘 1 𝑘 superscript 𝑧 1 𝑘 superscript 𝑧 1 𝑚 Euler-Gamma 𝑎 Euler-Gamma 𝑏 superscript subscript 𝑘 0 subscript 𝑎 𝑚 𝑘 subscript 𝑏 𝑚 𝑘 𝑘 𝑘 𝑚 superscript 1 𝑧 𝑘 1 𝑧 digamma 𝑘 1 digamma 𝑘 𝑚 1 digamma 𝑎 𝑘 𝑚 digamma 𝑏 𝑘 𝑚 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{1}% {\Gamma\left(a+m\right)\Gamma\left(b+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(b)% _{k}(m-k-1)!}{k!}(z-1)^{k}-\frac{(z-1)^{m}}{\Gamma\left(a\right)\Gamma\left(b% \right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(b+m)_{k}}{k!(k+m)!}(1-z)^{k}\*\left% (\ln\left(1-z\right)-\psi\left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k% +m\right)+\psi\left(b+k+m\right)\right),}}
    0 references
    DLMF id
    DLMF:15.8.E10
    0 references
    constraint
    | z - 1 | < 1 , | ph ⁡ ( 1 - z ) | < π formulae-sequence 𝑧 1 1 phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    test
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2agdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1idec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1gdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1gdec
    0 references
    Q11658
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1bdec
    0 references
    Q11651
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1ddec
    0 references
     
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