DLMF:15.8.E5 (Q5062)

From DRMF
Revision as of 14:01, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:15.8.E5
No description defined

    Statements

    Defining formula
    sin ⁡ ( π ⁢ ( c - a - b ) ) π ⁢ 𝐅 ⁡ ( a , b c ; z ) = z - a Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( c - b ) ⁢ 𝐅 ⁡ ( a , a - c + 1 a + b - c + 1 ; 1 - 1 z ) - ( 1 - z ) c - a - b ⁢ z a - c Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ 𝐅 ⁡ ( c - a , 1 - a c - a - b + 1 ; 1 - 1 z ) , 𝜋 𝑐 𝑎 𝑏 𝜋 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 superscript 𝑧 𝑎 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑐 𝑏 scaled-hypergeometric-bold-F 𝑎 𝑎 𝑐 1 𝑎 𝑏 𝑐 1 1 1 𝑧 superscript 1 𝑧 𝑐 𝑎 𝑏 superscript 𝑧 𝑎 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑏 scaled-hypergeometric-bold-F 𝑐 𝑎 1 𝑎 𝑐 𝑎 𝑏 1 1 1 𝑧 {\displaystyle{\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}% \left({a,b\atop c};z\right)=\frac{z^{-a}}{\Gamma\left(c-a\right)\Gamma\left(c-% b\right)}\mathbf{F}\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)-\frac{(1-% z)^{c-a-b}z^{a-c}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c% -a,1-a\atop c-a-b+1};1-\frac{1}{z}\right),}}
    0 references
    DLMF id
    DLMF:15.8.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.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1addec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    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.m2addec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1ddec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1ddec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1ddec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1ddec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5062&oldid=42663"