DLMF:15.8.E26 (Q5083)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:15.8.E26
No description defined

    Statements

    Defining formula
    F ⁡ ( a , 1 - a c ; z ) = ( 1 - z ) c - 1 ⁢ Γ ⁡ ( c ) ⁢ Γ ⁡ ( 1 2 ) Γ ⁡ ( 1 2 ⁢ ( c - a + 1 ) ) ⁢ Γ ⁡ ( 1 2 ⁢ c + 1 2 ⁢ a ) ⁢ F ⁡ ( 1 2 ⁢ c - 1 2 ⁢ a , 1 2 ⁢ c + 1 2 ⁢ a - 1 2 1 2 ; ( 1 - 2 ⁢ z ) 2 ) + ( 1 - 2 ⁢ z ) ⁢ ( 1 - z ) c - 1 ⁢ Γ ⁡ ( c ) ⁢ Γ ⁡ ( - 1 2 ) Γ ⁡ ( 1 2 ⁢ c - 1 2 ⁢ a ) ⁢ Γ ⁡ ( 1 2 ⁢ ( c + a - 1 ) ) ⁢ F ⁡ ( 1 2 ⁢ c - 1 2 ⁢ a + 1 2 , 1 2 ⁢ c + 1 2 ⁢ a 3 2 ; ( 1 - 2 ⁢ z ) 2 ) , Gauss-hypergeometric-F 𝑎 1 𝑎 𝑐 𝑧 superscript 1 𝑧 𝑐 1 Euler-Gamma 𝑐 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑐 𝑎 1 Euler-Gamma 1 2 𝑐 1 2 𝑎 Gauss-hypergeometric-F 1 2 𝑐 1 2 𝑎 1 2 𝑐 1 2 𝑎 1 2 1 2 superscript 1 2 𝑧 2 1 2 𝑧 superscript 1 𝑧 𝑐 1 Euler-Gamma 𝑐 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑐 1 2 𝑎 Euler-Gamma 1 2 𝑐 𝑎 1 Gauss-hypergeometric-F 1 2 𝑐 1 2 𝑎 1 2 1 2 𝑐 1 2 𝑎 3 2 superscript 1 2 𝑧 2 {\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}\frac{% \Gamma\left(c\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}(% c-a+1)\right)\Gamma\left(\tfrac{1}{2}c+\tfrac{1}{2}a\right)}F\left({\tfrac{1}{% 2}c-\tfrac{1}{2}a,\tfrac{1}{2}c+\tfrac{1}{2}a-\tfrac{1}{2}\atop\tfrac{1}{2}};(% 1-2z)^{2}\right)+(1-2z)(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(-% \tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}c-\tfrac{1}{2}a\right)\Gamma\left% (\tfrac{1}{2}(c+a-1)\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a+\tfrac{1}{2},% \tfrac{1}{2}c+\tfrac{1}{2}a\atop\tfrac{3}{2}};(1-2z)^{2}\right),}}
    0 references
    DLMF id
    DLMF:15.8.E26
    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.m2ajdec
    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.m2aodec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2apdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1apdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1ydec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1xdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1mdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5083&oldid=43537"