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DLMF:15.6.E8 - MaRDI portal

DLMF:15.6.E8 (Q5743)

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DLMF:15.6.E8
No description defined

    Statements

    DLMF defining formula
    𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( c - d ) ⁒ ∫ 0 1 𝐅 ⁑ ( a , b ; d ; z ⁒ t ) ⁒ t d - 1 ⁒ ( 1 - t ) c - d - 1 ⁒ d t , scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 1 Euler-Gamma 𝑐 𝑑 superscript subscript 0 1 scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑑 𝑧 𝑑 superscript 𝑑 𝑑 1 superscript 1 𝑑 𝑐 𝑑 1 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma% \left(c-d\right)}\int_{0}^{1}\mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-% 1}\mathrm{d}t,}}
    0 references
    constraint
    | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    β„œ ⁑ c > β„œ ⁑ d > 0 𝑐 𝑑 0 {\displaystyle{\displaystyle\Re c>\Re d>0}}
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    Symbols used
    gamma function
    DLMF defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    differential of x
    DLMF defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
    phase
    DLMF defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
    0 references
    real part
    DLMF defining formula
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
    Olver’s hypergeometric function
    DLMF 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.m2agdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1gdec
    0 references
    real or complex parameter
    DLMF defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1gdec
    0 references
    real or complex parameter
    DLMF defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1gdec
    0 references
    real or complex parameter
    DLMF defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1gdec
    0 references

    Identifiers

     
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