DLMF:15.10.E16 (Q5132)

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DLMF:15.10.E16
No description defined

    Statements

    Defining formula
    w 6 ⁢ ( z ) = e b ⁢ π ⁢ i ⁢ z - b ⁢ F ⁡ ( b , b - c + 1 b - a + 1 ; 1 z ) = e ( c - a ) ⁢ π ⁢ i ⁢ z a - c ⁢ ( 1 - z ) c - a - b ⁢ F ⁡ ( 1 - a , c - a b - a + 1 ; 1 z ) = ( 1 - z ) - b ⁢ F ⁡ ( b , c - a b - a + 1 ; 1 1 - z ) = e ( c - 1 ) ⁢ π ⁢ i ⁢ z 1 - c ⁢ ( 1 - z ) c - b - 1 ⁢ F ⁡ ( 1 - a , b - c + 1 b - a + 1 ; 1 1 - z ) . subscript 𝑤 6 𝑧 superscript 𝑒 𝑏 𝜋 imaginary-unit superscript 𝑧 𝑏 Gauss-hypergeometric-F 𝑏 𝑏 𝑐 1 𝑏 𝑎 1 1 𝑧 superscript 𝑒 𝑐 𝑎 𝜋 imaginary-unit superscript 𝑧 𝑎 𝑐 superscript 1 𝑧 𝑐 𝑎 𝑏 Gauss-hypergeometric-F 1 𝑎 𝑐 𝑎 𝑏 𝑎 1 1 𝑧 superscript 1 𝑧 𝑏 Gauss-hypergeometric-F 𝑏 𝑐 𝑎 𝑏 𝑎 1 1 1 𝑧 superscript 𝑒 𝑐 1 𝜋 imaginary-unit superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑏 1 Gauss-hypergeometric-F 1 𝑎 𝑏 𝑐 1 𝑏 𝑎 1 1 1 𝑧 {\displaystyle{\displaystyle w_{6}(z)=e^{b\pi\mathrm{i}}z^{-b}F\left({b,b-c+1% \atop b-a+1};\frac{1}{z}\right)=e^{(c-a)\pi\mathrm{i}}z^{a-c}(1-z)^{c-a-b}\*F% \left({1-a,c-a\atop b-a+1};\frac{1}{z}\right)=(1-z)^{-b}F\left({b,c-a\atop b-a% +1};\frac{1}{1-z}\right)=e^{(c-1)\pi\mathrm{i}}z^{1-c}(1-z)^{c-b-1}\*F\left({1% -a,b-c+1\atop b-a+1};\frac{1}{1-z}\right).}}
    0 references
    DLMF id
    DLMF:15.10.E16
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2akdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1odec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1odec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1odec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1ldec
    0 references
    Q11667
    Defining formula
    w m ⁢ ( z ) subscript 𝑤 𝑚 𝑧 {\displaystyle{\displaystyle w_{m}(z)}}
    xml-id
    C15.S10.XMD2.m1edec
    0 references
     
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