DLMF:15.16.E5 (Q5182): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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
 
Defining formula:

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.m2adec

Revision as of 14:27, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:15.16.E5
No description defined

    Statements

    Defining formula
    F ⁡ ( 1 2 + λ , - 1 2 - ν 1 + λ + μ ; z ) ⁢ F ⁡ ( 1 2 - λ , 1 2 + ν 1 + ν + μ ; 1 - z ) + F ⁡ ( 1 2 + λ , 1 2 - ν 1 + λ + μ ; z ) ⁢ F ⁡ ( - 1 2 - λ , 1 2 + ν 1 + ν + μ ; 1 - z ) - F ⁡ ( 1 2 + λ , 1 2 - ν 1 + λ + μ ; z ) ⁢ F ⁡ ( 1 2 - λ , 1 2 + ν 1 + ν + μ ; 1 - z ) = Γ ⁡ ( 1 + λ + μ ) ⁢ Γ ⁡ ( 1 + ν + μ ) Γ ⁡ ( λ + μ + ν + 3 2 ) ⁢ Γ ⁡ ( 1 2 + ν ) , Gauss-hypergeometric-F 1 2 𝜆 1 2 𝜈 1 𝜆 𝜇 𝑧 Gauss-hypergeometric-F 1 2 𝜆 1 2 𝜈 1 𝜈 𝜇 1 𝑧 Gauss-hypergeometric-F 1 2 𝜆 1 2 𝜈 1 𝜆 𝜇 𝑧 Gauss-hypergeometric-F 1 2 𝜆 1 2 𝜈 1 𝜈 𝜇 1 𝑧 Gauss-hypergeometric-F 1 2 𝜆 1 2 𝜈 1 𝜆 𝜇 𝑧 Gauss-hypergeometric-F 1 2 𝜆 1 2 𝜈 1 𝜈 𝜇 1 𝑧 Euler-Gamma 1 𝜆 𝜇 Euler-Gamma 1 𝜈 𝜇 Euler-Gamma 𝜆 𝜇 𝜈 3 2 Euler-Gamma 1 2 𝜈 {\displaystyle{\displaystyle F\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1% +\lambda+\mu};z\right)F\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+% \mu};1-z\right)+F\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu% };z\right)F\left({-\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z% \right)-F\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z% \right)F\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=% \frac{\Gamma\left(1+\lambda+\mu\right)\Gamma\left(1+\nu+\mu\right)}{\Gamma% \left(\lambda+\mu+\nu+\frac{3}{2}\right)\Gamma\left(\frac{1}{2}+\nu\right)},}}
    0 references
    DLMF id
    DLMF:15.16.E5
    0 references
    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
    0 references
    | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
    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.m2adec
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5182&oldid=47365"