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variants of this functions
MeijerG






Mathematica Notation

Traditional Notation









Hypergeometric Functions > MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z] > Integration > Definite integration > Integrals for classical integral transforms





http://functions.wolfram.com/07.34.21.0088.01









  


  










Input Form





Integrate[(\[Tau]^(\[Alpha] - 1) 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]^(l/k)])/E^(\[Sigma] \[Tau]), {\[Tau], 0, Infinity}] == ((k^\[Mu] l^(\[Alpha] - 1/2))/(\[Sigma]^\[Alpha] (2 Pi)^(SuperStar[c] (k - 1) + (l - 1)/2))) MeijerG[{{(1 - \[Alpha])/l, \[Ellipsis], (l - \[Alpha])/l, Subscript[a, 1]/k, \[Ellipsis], (Subscript[a, 1] + k - 1)/k, \[Ellipsis], Subscript[a, n]/k, \[Ellipsis], (Subscript[a, n] + k - 1)/ k}, {Subscript[a, n + 1]/k, \[Ellipsis], (Subscript[a, n + 1] + k - 1)/ k, \[Ellipsis], Subscript[a, p]/k, \[Ellipsis], (Subscript[a, p] + k - 1)/k}}, {{Subscript[b, 1]/k, \[Ellipsis], (Subscript[b, 1] + k - 1)/k, \[Ellipsis], Subscript[b, m]/k, \[Ellipsis], (Subscript[b, m] + k - 1)/k}, {Subscript[b, m + 1]/k, \[Ellipsis], (Subscript[b, m + 1] + k - 1)/k, \[Ellipsis], Subscript[b, q]/k, \[Ellipsis], (Subscript[b, q] + k - 1)/k}}, (\[Omega]^k l^l)/(\[Sigma]^l k^(k (q - p)))] /; Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 && GCD[k, l] == 1 && (\[DoubleStruckCapitalC]\[DoubleStruckCapitalC] /; s == v == 1 && t == u == Subscript[d, 1] == 0)










Standard Form





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MathML Form







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TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;-&quot;, &quot;1&quot;]], &quot;k&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> l </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8450;&#8450; </mi> <mo> /; </mo> <mrow> <mrow> <mi> s </mi> <mo> &#10869; </mo> <mi> v </mi> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> t </mi> <mo> &#10869; </mo> <mi> u </mi> <mo> &#10869; </mo> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["\[Tau]_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Sigma]_"]], " ", "\[Tau]_"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "n_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a_", RowBox[List["n_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "m_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", RowBox[List["m_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]]]], "}"]], ",", RowBox[List["\[Omega]_", " ", SuperscriptBox["\[Tau]_", FractionBox["l_", "k"]]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "\[Tau]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["k", "\[Mu]"], " ", SuperscriptBox["l", RowBox[List["\[Alpha]", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[Sigma]", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "l"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["l", "-", "\[Alpha]"]], "l"], ",", FractionBox[SubscriptBox["aa", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["aa", "n"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "n"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["aa", "p"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "p"], "+", "k", "-", "1"]], "k"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["bb", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "m"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "m"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "q"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "q"], "+", "k", "-", "1"]], "k"]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["\[Omega]", "k"], " ", SuperscriptBox["l", "l"]]], RowBox[List[SuperscriptBox["\[Sigma]", "l"], " ", SuperscriptBox["k", RowBox[List["k", " ", RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]]]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List[RowBox[List[SuperscriptBox["c", "*"], " ", RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]], "+", FractionBox[RowBox[List["l", "-", "1"]], "2"]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]], "&&", RowBox[List["l", "\[Element]", "Integers"]], "&&", RowBox[List["l", ">", "0"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "l"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["(", RowBox[List["\[DoubleStruckCapitalC]\[DoubleStruckCapitalC]", "/;", RowBox[List[RowBox[List["s", "\[Equal]", "v", "\[Equal]", "1"]], "&&", RowBox[List["t", "\[Equal]", "u", "\[Equal]", SubscriptBox["d", "1"], "\[Equal]", "0"]]]]]], ")"]]]]]]]]]]










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





2001-10-29