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/07.34.04.0011.01

 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, p]}}, x - I \[Epsilon]], \[Epsilon] -> Plus[0]] == Pi^(n - 1) Sum[((Product[Gamma[1 + Subscript[b, j] - Subscript[a, k]], {j, 1, m}]/Product[If[j == k, 1, Sin[Pi (Subscript[a, k] - Subscript[a, j])]] Product[Gamma[Subscript[a, k] - Subscript[b, j]], {j, m + 1, p}], {j, 1, n}]) x^(Subscript[a, k] - 1) HypergeometricPFQRegularized[ {1 + Subscript[b, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[b, p] - Subscript[a, k]}, {1 + Subscript[a, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, k - 1] - Subscript[a, k], 1 + Subscript[a, k + 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[a, k]}, (-1)^(p - m - n)/x] UnitStep[Abs[x] - 1])/ E^(2 I Pi Subscript[a, k]), {k, 1, n}] + 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, p]}, (-1)^(p - m - n) x] UnitStep[1 - Abs[x]])/ E^(2 I Pi Subscript[b, k]), {k, 1, m}] /; m + n - p <= 0 && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= n && 1 <= k <= n, !Element[Subscript[a, j] - Subscript[a, 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", "p"]]], "}"]]]], "}"]], ",", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", 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 MathML Form

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

 Rule Form

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

 2001-10-29