DLMF:15.6.E5 (Q5043)

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DLMF:15.6.E5
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    Statements

    test
    𝐅 ⁡ ( a , b ; c ; z ) = e - c ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 - b ) ⁢ Γ ⁡ ( 1 + b - c ) ⁢ 1 4 ⁢ π 2 ⁢ ∫ A ( 0 + , 1 + , 0 - , 1 - ) t b - 1 ⁢ ( 1 - t ) c - b - 1 ( 1 - z ⁢ t ) a ⁢ d t , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 superscript 𝑒 𝑐 𝜋 imaginary-unit Euler-Gamma 1 𝑏 Euler-Gamma 1 𝑏 𝑐 1 4 superscript 𝜋 2 superscript subscript 𝐴 limit-from 0 limit-from 1 limit-from 0 limit-from 1 superscript 𝑡 𝑏 1 superscript 1 𝑡 𝑐 𝑏 1 superscript 1 𝑧 𝑡 𝑎 𝑡 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-c\pi\mathrm{i}}% \Gamma\left(1-b\right)\Gamma\left(1+b-c\right)\*\frac{1}{4\pi^{2}}\int_{A}^{(0% +,1+,0-,1-)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,}}
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    DLMF id
    DLMF:15.6.E5
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    constraint
    | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    | ph ⁡ ( 1 - z ) | < π ph 1 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    b , c - b ≠ 1 , 2 , 3 , … formulae-sequence 𝑏 𝑐 𝑏 1 2 3 … {\displaystyle{\displaystyle b,c-b\neq 1,2,3,\dots}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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