DLMF:16.17.E1 (Q5277): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: Q11690 / rank
 
Normal rank
Property / Symbols used: Q11690 / qualifier
 
Defining formula:

p 𝑝 {\displaystyle{\displaystyle p}}

p
Property / Symbols used: Q11690 / qualifier
 
xml-id: C16.S1.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q11691 / rank
 
Normal rank
Property / Symbols used: Q11691 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11691 / qualifier
 
xml-id: C16.S1.XMD2.m1dec
Property / Symbols used
 
Property / Symbols used: Q11692 / rank
 
Normal rank
Property / Symbols used: Q11692 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11692 / qualifier
 
xml-id: C16.S1.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q11693 / rank
 
Normal rank
Property / Symbols used: Q11693 / qualifier
 
Defining formula:

a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}

a,a_{1},\ldots,a_{p}
Property / Symbols used: Q11693 / qualifier
 
xml-id: C16.S1.XMD4.m1dec
Property / Symbols used
 
Property / Symbols used: Q11694 / rank
 
Normal rank
Property / Symbols used: Q11694 / qualifier
 
Defining formula:

b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}

b,b_{1},\ldots,b_{q}
Property / Symbols used: Q11694 / qualifier
 
xml-id: C16.S1.XMD5.m1dec

Latest revision as of 14:42, 2 January 2020

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DLMF:16.17.E1
No description defined

    Statements

    Defining formula
    G p , q m , n ⁡ ( z ; 𝐚 ; 𝐛 ) = G p , q m , n ⁡ ( z ; a 1 , … , a p b 1 , … , b q ) = 1 2 ⁢ π ⁢ i ⁢ ∫ L ( ∏ ℓ = 1 m Γ ⁡ ( b ℓ - s ) ⁢ ∏ ℓ = 1 n Γ ⁡ ( 1 - a ℓ + s ) / ( ∏ ℓ = m q - 1 Γ ⁡ ( 1 - b ℓ + 1 + s ) ⁢ ∏ ℓ = n p - 1 Γ ⁡ ( a ℓ + 1 - s ) ) ) ⁢ z s ⁢ d s , Meijer-G 𝑚 𝑛 𝑝 𝑞 𝑧 𝐚 𝐛 Meijer-G 𝑚 𝑛 𝑝 𝑞 𝑧 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 1 2 𝜋 imaginary-unit subscript 𝐿 superscript subscript product ℓ 1 𝑚 Euler-Gamma subscript 𝑏 ℓ 𝑠 superscript subscript product ℓ 1 𝑛 Euler-Gamma 1 subscript 𝑎 ℓ 𝑠 superscript subscript product ℓ 𝑚 𝑞 1 Euler-Gamma 1 subscript 𝑏 ℓ 1 𝑠 superscript subscript product ℓ 𝑛 𝑝 1 Euler-Gamma subscript 𝑎 ℓ 1 𝑠 superscript 𝑧 𝑠 𝑠 {\displaystyle{\displaystyle{G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right% )={G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)=% \frac{1}{2\pi\mathrm{i}}\int_{L}\left({\textstyle\ifrac{\prod\limits_{\ell=1}^% {m}\Gamma\left(b_{\ell}-s\right)\prod\limits_{\ell=1}^{n}\Gamma\left(1-a_{\ell% }+s\right)}{\left(\prod\limits_{\ell=m}^{q-1}\Gamma\left(1-b_{\ell+1}+s\right)% \prod\limits_{\ell=n}^{p-1}\Gamma\left(a_{\ell+1}-s\right)\right)}}\right)z^{s% }\mathrm{d}s,}}
    0 references
    DLMF id
    DLMF:16.17.E1
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    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
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q11690
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1dec
    0 references
    Q11691
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1dec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1dec
    0 references
    Q11693
    Defining formula
    a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
    xml-id
    C16.S1.XMD4.m1dec
    0 references
    Q11694
    Defining formula
    b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}
    xml-id
    C16.S1.XMD5.m1dec
    0 references
     
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