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 MeijerG

 http://functions.wolfram.com/07.34.04.0013.01

 Input Form

 Limit[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]}}, x - I \[Epsilon]], \[Epsilon] -> Plus[0]] == Sum[(Product[If[j == k, 1, Gamma[Subscript[b, j] - Subscript[b, k]]] Product[Gamma[1 + Subscript[b, k] - Subscript[a, j]], {j, 1, n}], {j, 1, m}]/Product[Gamma[Subscript[a, j] - Subscript[b, k]] Product[Gamma[1 + Subscript[b, k] - Subscript[b, j]], {j, m + 1, q}], {j, n + 1, p}]) x^Subscript[b, k] Exp[-2 Subscript[b, k] Pi I] HypergeometricPFQ[{1 + Subscript[b, k] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[a, p]}, {1 + Subscript[b, k] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, k - 1], 1 + Subscript[b, k] - Subscript[b, k + 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, q]}, (-1)^(p - m - n) x], {k, 1, m}] /; (p < q || (p == q && m + n > p) || (p == q && m + n == p && Abs[x] < 1)) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= m && 1 <= k <= m, !Element[Subscript[b, j] - Subscript[b, k], Integers]] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[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["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "]"]]]], "]"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "j"]]], "]"]]]], ")"]], " "]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], "p"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["b", "k"]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["m", "+", "1"]]]], "q"], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", "j"]]], "]"]]]]]]]]], " ", SuperscriptBox["x", SubscriptBox["b", "k"]], RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "2"]], SubscriptBox["b", "k"], " ", "\[Pi]", " ", "\[ImaginaryI]"]], " ", "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "p"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]]]], ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "x"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["p", "<", "q"]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["m", "+", "n"]], ">", "p"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["m", "+", "n"]], "\[Equal]", "p"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "x", "]"]], "<", "1"]]]], ")"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[NotEqual]", "k"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "m"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], ")"]]]], ")"]]]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 lim ϵ "\[Rule]" + 0 TagBox["", HypergeometricPFQ] G TagBox["G", MeijerG] p , q m , n ( x - ϵ TagBox[RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], MeijerG, Rule[Editable, True]] a 1 TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , a n TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]] , a n + 1 TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , a p TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]] b 1 TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , b m TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]] , b m + 1 TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , b q TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]] ) k = 1 m ( j = 1 j k m Γ ( b j - b k ) ) j = 1 n Γ ( 1 - a j + b k ) j = n + 1 p Γ ( a j - b k ) j = m + 1 q Γ ( 1 - b j + b k ) x b k - 2 b k π p F q - 1 ( 1 - a 1 + b k , , 1 - a p + b k TagBox[TagBox[TagBox[RowBox[List[RowBox[List["1", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; 1 - b 1 + b k , , 1 - b k - 1 + b k , 1 - b k + 1 + b k , , 1 - b q + b k TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List["1", "-", SubscriptBox["b", "q"], "+", TagBox[SubscriptBox["b", "k"], HypergeometricPFQ, Rule[Editable, True]]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; ( - 1 ) p - m - n x TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "x"]], HypergeometricPFQRegularized, Rule[Editable, True]] ) /; ( p < q p q m + n > p p q m + n p "\[LeftBracketingBar]" x "\[RightBracketingBar]" < 1 ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j m 1 k m ( b j - b k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) x < 0 FormBox RowBox RowBox RowBox Limit [ RowBox TagBox HypergeometricPFQ RowBox SubsuperscriptBox TagBox G MeijerG RowBox p , q RowBox m , n RowBox ( RowBox TagBox RowBox x - RowBox ϵ MeijerG Rule Editable GridBox RowBox TagBox SubscriptBox a 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox a n MeijerG Rule Editable , TagBox SubscriptBox a RowBox n + 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox a p MeijerG Rule Editable RowBox TagBox SubscriptBox b 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox b m MeijerG Rule Editable , TagBox SubscriptBox b RowBox m + 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox b q MeijerG Rule Editable ) , RowBox ϵ -> RowBox + 0 ] RowBox UnderoverscriptBox RowBox k = 1 m RowBox FractionBox RowBox RowBox ( RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k m RowBox Γ ( RowBox SubscriptBox b j - SubscriptBox b k ) ) RowBox UnderoverscriptBox RowBox j = 1 n RowBox Γ ( RowBox 1 - SubscriptBox a j + SubscriptBox b k ) RowBox UnderoverscriptBox RowBox j = RowBox n + 1 p RowBox RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox b k ) RowBox UnderoverscriptBox RowBox j = RowBox m + 1 q RowBox Γ ( RowBox 1 - SubscriptBox b j + SubscriptBox b k ) SuperscriptBox x SubscriptBox b k SuperscriptBox RowBox RowBox - 2 SubscriptBox b k π RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox F FormBox RowBox q - 1 TraditionalForm RowBox ( RowBox TagBox TagBox TagBox RowBox RowBox 1 TagBox RowBox RowBox - SubscriptBox a 1 + SubscriptBox b k HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a p + SubscriptBox b k HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox TagBox RowBox 1 - SubscriptBox b 1 + SubscriptBox b k HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k - 1 + SubscriptBox b k HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k + 1 + SubscriptBox b k HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , RowBox 1 - SubscriptBox b q + TagBox SubscriptBox b k HypergeometricPFQ Rule Editable HypergeometricPFQRegularized Rule Editable ; TagBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox p - m - n x HypergeometricPFQRegularized Rule Editable ) /; RowBox RowBox ( RowBox RowBox p < q RowBox RowBox p q RowBox RowBox m + n > p RowBox RowBox p q RowBox RowBox m + n p RowBox RowBox x < 1 ) RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox 1 j m RowBox 1 k m RowBox ( RowBox RowBox SubscriptBox b j - SubscriptBox b k TagBox Function ) RowBox x < 0 TraditionalForm [/itex]

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

 2007-05-02