Meijer G-function: Integration (formula 07.34.21.0095)

MeijerG

$Mathematica Notation$

Hypergeometric Functions

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z]

Integration

Definite integration

Integrals including G functions with rational arguments

http://functions.wolfram.com/07.34.21.0095.01

Input Form

Integrate[(\[Tau]^(\[Alpha] - 1)/(\[Tau] + z)^\[Beta]) MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, (\[Omega] (\[Tau] + z)^l)/\[Tau]^k], {\[Tau], 0, Infinity}] == ((Sqrt[2 Pi] k^(\[Alpha] - 1/2) l^(1/2 - \[Beta]) z^(\[Alpha] - \[Beta]))/ (l - k)^(\[Alpha] - \[Beta] + 1/2)) MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p], \[Beta]/l, \[Ellipsis], (\[Beta] + l - 1)/l}}, {{\[Alpha]/k, \[Ellipsis], (\[Alpha] + k - 1)/k, (\[Beta] - \[Alpha])/(l - k), \[Ellipsis], (\[Beta] - \[Alpha] + l - k - 1)/(l - k), Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, ((\[Omega] l^l)/k^k) (z/(l - k))^(l - k)] /; Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 && k < l && SuperStar[c] > 0 && Abs[Arg[\[Omega] z^(l - k)]] < Pi SuperStar[c] && (l - k) (1 - Max[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, n]]]) + Re[\[Beta] - \[Alpha]] > 0 && k - k Max[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, n]]] + Re[\[Alpha]] > 0

Standard Form

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MathML Form

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</mo> <mi> k </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> l </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> , </mo> <mfrac> <mi> β </mi> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> β </mi> <mo> + </mo> <mi> l </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> l </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mi> α </mi> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> β </mi> <mo> - </mo> <mi> α </mi> </mrow> <mrow> <mi> l </mi> <mo> - 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</mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z,r]