DLMF:15.6.E4 (Q5042)

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DLMF:15.6.E4
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    Statements

    Defining formula
    𝐅 ⁑ ( a , b ; c ; z ) = e - b ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( c - b ) ⁒ ∫ 1 ( 0 + ) t b - 1 ⁒ ( 1 - t ) c - b - 1 ( 1 - z ⁒ t ) a ⁒ d t , scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 𝑒 𝑏 πœ‹ imaginary-unit Euler-Gamma 1 𝑏 2 πœ‹ imaginary-unit Euler-Gamma 𝑐 𝑏 superscript subscript 1 limit-from 0 superscript 𝑑 𝑏 1 superscript 1 𝑑 𝑐 𝑏 1 superscript 1 𝑧 𝑑 π‘Ž 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}% \frac{\Gamma\left(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(% 0+)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,}}
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    DLMF id
    DLMF:15.6.E4
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    constraint
    | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    β„œ ⁑ ( c - b ) > 0 𝑐 𝑏 0 {\displaystyle{\displaystyle\Re(c-b)>0}}
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    | ph ⁑ ( 1 - z ) | < Ο€ ph 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    b β‰  1 , 2 , 3 , … 𝑏 1 2 3 … {\displaystyle{\displaystyle b\neq 1,2,3,\dots}}
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    β„œ ⁑ ( c - b ) > 0 𝑐 𝑏 0 {\displaystyle{\displaystyle\Re(c-b)>0}}
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    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
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    Q10811
    Defining formula
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2acdec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1cdec
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    Q11645
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1cdec
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