Meijer G-function: Integration (formula 07.34.21.0096)

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/07.34.21.0096.01

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]))/ (k - l)^(\[Alpha] - \[Beta] + 1/2)) MeijerG[{{(\[Alpha] - \[Beta] + 1)/(k - l), \[Ellipsis], (\[Alpha] - \[Beta] + k - l)/(k - l), 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, Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, ((\[Omega] l^l)/k^k) (z/(k - l))^(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] && (k - l) Min[Re[Subscript[b, 1]], \[Ellipsis], Re[Subscript[b, m]]] + 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> </msup> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mo> * </mo> </msup> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> β </mi> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> k </mi> <mo> - </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 encoding='MathML-Content'> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mfrac> <mrow> <msup> <mi> τ </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> τ </mi> </mrow> <mo> ) </mo> </mrow> <mi> β </mi> </msup> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ω </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> τ </mi> </mrow> <mo> ) </mo> </mrow> <mi> l </mi> </msup> </mrow> <msup> <mi> τ </mi> <mi> k </mi> </msup> </mfrac> <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> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["\[Omega]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "\[Tau]"]], ")"]], "l"]]], SuperscriptBox["\[Tau]", "k"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> τ </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msup> <mi> k </mi> <mrow> <mi> α </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> l </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mtext> </mtext> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> α </mi> <mo> - </mo> <mi> β </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> G </mi> <annotation encoding='Mathematica'> TagBox["G", MeijerG] </annotation> </semantics> <mrow> <mrow> <mi> p </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mi> q </mi> <mo> + </mo> <mi> k </mi> </mrow> </mrow> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mfrac> <mrow> <mi> ω </mi> <mo> ⁢ </mo> <msup> <mi> l </mi> <mi> l </mi> </msup> </mrow> <msup> <mi> k </mi> <mi> k </mi> </msup> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> l </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[FractionBox[RowBox[List["\[Omega]", " ", SuperscriptBox["l", "l"]]], SuperscriptBox["k", "k"]], SuperscriptBox[RowBox[List["(", FractionBox["z", RowBox[List["k", "-", "l"]]], ")"]], RowBox[List["l", "-", "k"]]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <semantics> <mfrac> <mrow> <mi> α </mi> <mo> - </mo> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List["\[Alpha]", "-", "\[Beta]", "+", "1"]], RowBox[List["k", "-", "l"]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mrow> <mi> α </mi> <mo> - </mo> <mi> β </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List["\[Alpha]", "-", "\[Beta]", "+", "k", "-", "l"]], RowBox[List["k", "-", "l"]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mi> p </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mi> β </mi> <mi> l </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox["\[Beta]", "l"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mrow> <mi> β </mi> <mo> + </mo> <mi> l </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> l </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List["\[Beta]", "+", "l", "-", "1"]], "l"], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <semantics> <mfrac> <mi> α </mi> <mi> k </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox["\[Alpha]", "k"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mi> q </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> l </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> k </mi> <mo> > </mo> <mi> l </mi> </mrow> <mo> ∧ </mo> <mrow> <msup> <mi> c </mi> <mo> * </mo> </msup> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ω </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> l </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mo> * </mo> </msup> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> β </mi> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> k </mi> <mo> - </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]