html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 MeijerG

 http://functions.wolfram.com/07.34.04.0008.02

 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]] == Pi^(m - 1) Sum[((Product[Gamma[1 + Subscript[b, k] - Subscript[a, j]], {j, 1, n}]/ Product[If[j == k, 1, Sin[Pi (Subscript[b, j] - Subscript[b, k])]] Product[Gamma[Subscript[a, j] - Subscript[b, k]], {j, n + 1, p}], {j, 1, m}]) x^Subscript[b, k] HypergeometricPFQRegularized[ {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])/E^(2 I Pi Subscript[b, k]), {k, 1, m}] /; (p < q || (p == q && m + n > p) || (p == q && m + n == p && Abs[z] < 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

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 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]] ) π m - 1 k = 1 m j = 1 n Γ ( 1 - a j + b k ) j = 1 j k m sin ( π ( b j - b k ) ) j = n + 1 p Γ ( a j - b k ) - 2 π b k x b k p F ~ q - 1 ( 1 - a 1 + b k , , 1 - a p + b k ; 1 - b 1 + b k , , 1 - b k - 1 + b k , 1 - b k + 1 + b k , , 1 - b q + b k ; ( - 1 ) p - m - n x ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["q", "-", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "x"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] /; ( p < q p q m + n > p p q m + n p "\[LeftBracketingBar]" z "\[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 SuperscriptBox π RowBox m - 1 RowBox UnderoverscriptBox RowBox k = 1 m RowBox FractionBox RowBox UnderoverscriptBox RowBox j = 1 n RowBox Γ ( RowBox 1 - SubscriptBox a j + SubscriptBox b k ) RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k m RowBox sin ( RowBox π RowBox ( RowBox SubscriptBox b j - SubscriptBox b k ) ) RowBox UnderoverscriptBox RowBox j = RowBox n + 1 p RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox b k ) SuperscriptBox RowBox RowBox - 2 π SubscriptBox b k SuperscriptBox x SubscriptBox b k TagBox TagBox RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox OverscriptBox F ~ FormBox RowBox q - 1 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox 1 - SubscriptBox a 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - SubscriptBox a p + SubscriptBox b k HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox TagBox RowBox TagBox RowBox 1 - SubscriptBox b 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k - 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k + 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox TagBox RowBox 1 - SubscriptBox b q + SubscriptBox b k HypergeometricPFQRegularized Rule Editable HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox p - m - n x HypergeometricPFQRegularized Rule Editable ) InterpretTemplate Function HypergeometricPFQRegularized Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQRegularized /; 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 z < 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]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["m", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "j"]]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["b", "k"]]]], " ", SuperscriptBox["x", SubscriptBox["b", "k"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["aa", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["aa", "p"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["bb", "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["bb", "q"]]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "x"]]]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], ")"]]]], "]"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], "p"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["b", "k"]]], "]"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["p", "<", "q"]], "||", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "&&", RowBox[List[RowBox[List["m", "+", "n"]], ">", "p"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "&&", RowBox[List[RowBox[List["m", "+", "n"]], "\[Equal]", "p"]], "&&", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]], ")"]]]], ")"]], "&&", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[NotEqual]", "k"]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "m"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "m"]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]]]], ")"]]]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

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

 2001-10-29