DLMF:15.10.E15 (Q5131)

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DLMF:15.10.E15
No description defined

    Statements

    Defining formula
    w 5 ⁢ ( z ) = e a ⁢ π ⁢ i ⁢ z - a ⁢ F ⁡ ( a , a - c + 1 a - b + 1 ; 1 z ) = e ( c - b ) ⁢ π ⁢ i ⁢ z b - c ⁢ ( 1 - z ) c - a - b ⁢ F ⁡ ( 1 - b , c - b a - b + 1 ; 1 z ) = ( 1 - z ) - a ⁢ F ⁡ ( a , c - b a - b + 1 ; 1 1 - z ) = e ( c - 1 ) ⁢ π ⁢ i ⁢ z 1 - c ⁢ ( 1 - z ) c - a - 1 ⁢ F ⁡ ( 1 - b , a - c + 1 a - b + 1 ; 1 1 - z ) . subscript 𝑤 5 𝑧 superscript 𝑒 𝑎 𝜋 imaginary-unit superscript 𝑧 𝑎 Gauss-hypergeometric-F 𝑎 𝑎 𝑐 1 𝑎 𝑏 1 1 𝑧 superscript 𝑒 𝑐 𝑏 𝜋 imaginary-unit superscript 𝑧 𝑏 𝑐 superscript 1 𝑧 𝑐 𝑎 𝑏 Gauss-hypergeometric-F 1 𝑏 𝑐 𝑏 𝑎 𝑏 1 1 𝑧 superscript 1 𝑧 𝑎 Gauss-hypergeometric-F 𝑎 𝑐 𝑏 𝑎 𝑏 1 1 1 𝑧 superscript 𝑒 𝑐 1 𝜋 imaginary-unit superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑎 1 Gauss-hypergeometric-F 1 𝑏 𝑎 𝑐 1 𝑎 𝑏 1 1 1 𝑧 {\displaystyle{\displaystyle w_{5}(z)=e^{a\pi\mathrm{i}}z^{-a}\*F\left({a,a-c+% 1\atop a-b+1};\frac{1}{z}\right)=e^{(c-b)\pi\mathrm{i}}z^{b-c}(1-z)^{c-a-b}\*F% \left({1-b,c-b\atop a-b+1};\frac{1}{z}\right)=(1-z)^{-a}F\left({a,c-b\atop a-b% +1};\frac{1}{1-z}\right)=e^{(c-1)\pi\mathrm{i}}z^{1-c}(1-z)^{c-a-1}\*F\left({1% -b,a-c+1\atop a-b+1};\frac{1}{1-z}\right).}}
    0 references
    DLMF id
    DLMF:15.10.E15
    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.m2ajdec
    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
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1ndec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1ndec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1ndec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1kdec
    0 references
    Q11667
    Defining formula
    w m ⁢ ( z ) subscript 𝑤 𝑚 𝑧 {\displaystyle{\displaystyle w_{m}(z)}}
    xml-id
    C15.S10.XMD2.m1ddec
    0 references
     
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