DLMF:15.8.E8 (Q5065): Difference between revisions

From testwiki
Jump to navigation Jump to search
← Older edit
imported>Admin
imported>Admin
 
(9 intermediate revisions by the same user not shown)
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.m2adec
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.m5adec
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.m2adec
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.m1aedec
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.m2aedec
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.m1gdec
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.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.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.m1dec
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.m1bdec

Latest revision as of 13:02, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:15.8.E8
No description defined

    Statements

    test
    𝐅 ⁡ ( a , a + m c ; z ) = ( - z ) - a Γ ⁡ ( a + m ) ⁢ ∑ k = 0 m - 1 ( a ) k ⁢ ( m - k - 1 ) ! k ! ⁢ Γ ⁡ ( c - a - k ) ⁢ z - k + ( - z ) - a Γ ⁡ ( a ) ⁢ ∑ k = 0 ∞ ( a + m ) k k ! ⁢ ( k + m ) ! ⁢ Γ ⁡ ( c - a - k - m ) ⁢ ( - 1 ) k ⁢ z - k - m ⁢ ( ln ⁡ ( - z ) + ψ ⁡ ( k + 1 ) + ψ ⁡ ( k + m + 1 ) - ψ ⁡ ( a + k + m ) - ψ ⁡ ( c - a - k - m ) ) , scaled-hypergeometric-bold-F 𝑎 𝑎 𝑚 𝑐 𝑧 superscript 𝑧 𝑎 Euler-Gamma 𝑎 𝑚 superscript subscript 𝑘 0 𝑚 1 subscript 𝑎 𝑘 𝑚 𝑘 1 𝑘 Euler-Gamma 𝑐 𝑎 𝑘 superscript 𝑧 𝑘 superscript 𝑧 𝑎 Euler-Gamma 𝑎 superscript subscript 𝑘 0 subscript 𝑎 𝑚 𝑘 𝑘 𝑘 𝑚 Euler-Gamma 𝑐 𝑎 𝑘 𝑚 superscript 1 𝑘 superscript 𝑧 𝑘 𝑚 𝑧 digamma 𝑘 1 digamma 𝑘 𝑚 1 digamma 𝑎 𝑘 𝑚 digamma 𝑐 𝑎 𝑘 𝑚 {\displaystyle{\displaystyle\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(-z)% ^{-a}}{\Gamma\left(a+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma% \left(c-a-k\right)}z^{-k}+\frac{(-z)^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{% \infty}\frac{(a+m)_{k}}{k!(k+m)!\Gamma\left(c-a-k-m\right)}(-1)^{k}z^{-k-m}\*% \left(\ln\left(-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left% (a+k+m\right)-\psi\left(c-a-k-m\right)\right),}}
    0 references
    DLMF id
    DLMF:15.8.E8
    0 references
    constraint
    | z | > 1 , | ph ⁡ ( - z ) | < π formulae-sequence 𝑧 1 phase 𝑧 𝜋 {\displaystyle{\displaystyle|z|>1,|\operatorname{ph}\left(-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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aedec
    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.m2aedec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1gdec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1edec
    0 references
    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1gdec
    0 references
    Q11658
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1dec
    0 references
    Q11651
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1bdec
    0 references
     
      Retrieved from "https://lct.wmflabs.org/w/index.php?title=Item:Q5065&oldid=28886"