DLMF:15.14.E1 (Q5176)

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DLMF:15.14.E1
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ x s - 1 ⁒ 𝐅 ⁑ ( a , b c ; - x ) ⁒ d x = Ξ“ ⁑ ( s ) ⁒ Ξ“ ⁑ ( a - s ) ⁒ Ξ“ ⁑ ( b - s ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c - s ) , superscript subscript 0 superscript π‘₯ 𝑠 1 scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 π‘₯ π‘₯ Euler-Gamma 𝑠 Euler-Gamma π‘Ž 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma π‘Ž Euler-Gamma 𝑏 Euler-Gamma 𝑐 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c% };-x\right)\mathrm{d}x=\frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma% \left(b-s\right)}{\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left(c-s% \right)},}}
    0 references
    DLMF id
    DLMF:15.14.E1
    0 references
    constraint
    min ⁑ ( β„œ ⁑ a , β„œ ⁑ b ) > β„œ ⁑ s > 0 π‘Ž 𝑏 𝑠 0 {\displaystyle{\displaystyle\min(\Re a,\Re b)>\Re s>0}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10811
    Defining formula
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2adec
    0 references
    Q11649
    Defining formula
    x π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C15.S1.XMD1.m1dec
    0 references
    Q11648
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C15.S1.XMD11.m1dec
    0 references
    Q11645
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1dec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1dec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1dec
    0 references
     
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