DLMF:31.10.E11 (Q9062): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(8 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: Q12457 / rank
 
Normal rank
Property / Symbols used: Q12457 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12457 / qualifier
 
xml-id: C31.S1.XMD1.m1edec
Property / Symbols used
 
Property / Symbols used: Q12458 / rank
 
Normal rank
Property / Symbols used: Q12458 / qualifier
 
Defining formula:

γ 𝛾 {\displaystyle{\displaystyle\gamma}}

\gamma
Property / Symbols used: Q12458 / qualifier
 
xml-id: C31.S1.XMD10.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12459 / rank
 
Normal rank
Property / Symbols used: Q12459 / qualifier
 
Defining formula:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q12459 / qualifier
 
xml-id: C31.S1.XMD11.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12460 / rank
 
Normal rank
Property / Symbols used: Q12460 / qualifier
 
Defining formula:

ϵ italic-ϵ {\displaystyle{\displaystyle\epsilon}}

\epsilon
Property / Symbols used: Q12460 / qualifier
 
xml-id: C31.S1.XMD12.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12461 / rank
 
Normal rank
Property / Symbols used: Q12461 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12461 / qualifier
 
xml-id: C31.S1.XMD6.m1edec
Property / Symbols used
 
Property / Symbols used: Q12463 / rank
 
Normal rank
Property / Symbols used: Q12463 / qualifier
 
Defining formula:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q12463 / qualifier
 
xml-id: C31.S1.XMD8.m1ddec
Property / Symbols used
 
Property / Symbols used: Q12464 / rank
 
Normal rank
Property / Symbols used: Q12464 / qualifier
 
Defining formula:

β 𝛽 {\displaystyle{\displaystyle\beta}}

\beta
Property / Symbols used: Q12464 / qualifier
 
xml-id: C31.S1.XMD9.m1ddec
Property / Symbols used
 
Property / Symbols used: kernel (locally) / rank
 
Normal rank
Property / Symbols used: kernel (locally) / qualifier
 
Defining formula:

𝒦 ⁢ ( z , t ) 𝒦 𝑧 𝑡 {\displaystyle{\displaystyle\mathcal{K}(z,t)}}

\mathcal{K}(z,t)
Property / Symbols used: kernel (locally) / qualifier
 
xml-id: C31.S10.XMD4.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12497 / rank
 
Normal rank
Property / Symbols used: Q12497 / qualifier
 
Defining formula:

σ 𝜎 {\displaystyle{\displaystyle\sigma}}

\sigma
Property / Symbols used: Q12497 / qualifier
 
xml-id: C31.S10.XMD8.m1bdec

Latest revision as of 13:37, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:31.10.E11
No description defined

    Statements

    Defining formula
    𝒦 ⁢ ( z , t ) = ( z ⁢ t - a ) 1 2 - δ - σ ⁢ ( z ⁢ t / a ) - 1 2 + δ + σ - α ⁢ F 1 2 ⁡ ( 1 2 - δ - σ + α , 3 2 - δ - σ + α - γ α - β + 1 ; a z ⁢ t ) ⁢ P ⁡ { 0 1 ∞ 0 0 - 1 2 + δ + σ ( z - a ) ⁢ ( t - a ) ( 1 - a ) ⁢ ( z ⁢ t - a ) 1 - ϵ 1 - δ - 1 2 + ϵ - σ } . 𝒦 𝑧 𝑡 superscript 𝑧 𝑡 𝑎 1 2 𝛿 𝜎 superscript 𝑧 𝑡 𝑎 1 2 𝛿 𝜎 𝛼 Gauss-hypergeometric-F-as-2F1 1 2 𝛿 𝜎 𝛼 3 2 𝛿 𝜎 𝛼 𝛾 𝛼 𝛽 1 𝑎 𝑧 𝑡 Riemann-P-symbol 0 1 absent 0 0 1 2 𝛿 𝜎 𝑧 𝑎 𝑡 𝑎 1 𝑎 𝑧 𝑡 𝑎 1 italic-ϵ 1 𝛿 1 2 italic-ϵ 𝜎 absent {\displaystyle{\displaystyle\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma% }\left(\ifrac{zt}{a}\right)^{-\frac{1}{2}+\delta+\sigma-\alpha}\*{{}_{2}F_{1}}% \left({\frac{1}{2}-\delta-\sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-% \gamma\atop\alpha-\beta+1};\frac{a}{zt}\right)\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.}}
    0 references
    DLMF id
    DLMF:31.10.E11
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5aadec
    0 references
    Riemann’s $$P$$ -symbol for solutions of the generalized hypergeometric differential equation
    Defining formula
    P ⁡ { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } Riemann-P-symbol 𝛼 𝛽 𝛾 absent subscript 𝑎 1 subscript 𝑏 1 subscript 𝑐 1 𝑧 subscript 𝑎 2 subscript 𝑏 2 subscript 𝑐 2 absent {\displaystyle{\displaystyle P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}}}
    xml-id
    C15.S11.E3.m2aadec
    0 references
    Q12457
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C31.S1.XMD1.m1edec
    0 references
    Q12458
    Defining formula
    γ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C31.S1.XMD10.m1fdec
    0 references
    Q12459
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C31.S1.XMD11.m1fdec
    0 references
    Q12460
    Defining formula
    ϵ italic-ϵ {\displaystyle{\displaystyle\epsilon}}
    xml-id
    C31.S1.XMD12.m1fdec
    0 references
    Q12461
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C31.S1.XMD6.m1edec
    0 references
    Q12463
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C31.S1.XMD8.m1ddec
    0 references
    Q12464
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C31.S1.XMD9.m1ddec
    0 references
    kernel (locally)
    Defining formula
    𝒦 ⁢ ( z , t ) 𝒦 𝑧 𝑡 {\displaystyle{\displaystyle\mathcal{K}(z,t)}}
    xml-id
    C31.S10.XMD4.m1fdec
    0 references
    Q12497
    Defining formula
    σ 𝜎 {\displaystyle{\displaystyle\sigma}}
    xml-id
    C31.S10.XMD8.m1bdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q9062&oldid=38333"