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; }

 KelvinKer

 http://functions.wolfram.com/03.20.26.0044.01

 Input Form

 BesselK[\[Nu], (-1)^(1/4) z] KelvinKer[\[Nu], z] == ((E^((3 I Pi \[Nu])/4) Pi^(5/2) z^\[Nu] (-I + Cot[Pi \[Nu]]) Csc[Pi (3/4 + \[Nu])])/(((-1)^(1/4) z)^\[Nu] (4 Sqrt[2]))) MeijerG[{{1/2}, {1/4, -(1/4) + \[Nu]}}, {{0, \[Nu]}, {1/4, -\[Nu], -(1/4) + \[Nu]}}, I z^2] - ((Pi^(5/2) ((-1)^(1/4) z)^\[Nu] Csc[Pi \[Nu]])/(E^((3 I Pi \[Nu])/4) z^\[Nu] (4 Sqrt[2]))) Csc[Pi (\[Nu] + 1/4)] MeijerG[{{1/2}, {1/4, -\[Nu] - 1/4}}, {{0, -\[Nu]}, {\[Nu], 1/4, -\[Nu] - 1/4}}, I z^2] - ((1/16) Sqrt[Pi] z^\[Nu] (I + Cot[Pi \[Nu]]) MeijerG[{{}, {}}, {{0, 1/2, \[Nu]/2}, {-(\[Nu]/2)}}, -(z^4/64)])/ (E^((3/4) I Pi \[Nu]) ((-1)^(1/4) z)^\[Nu]) + ((E^((3 I Pi \[Nu])/4) Sqrt[Pi] ((-1)^(1/4) z)^\[Nu] Csc[Pi \[Nu]])/ (z^\[Nu] 16)) MeijerG[{{}, {}}, {{0, 1/2, -(\[Nu]/2)}, {\[Nu]/2}}, -(z^4/64)] /; !Element[\[Nu], Integers] && Inequality[-(Pi/2), Less, Arg[z], LessEqual, 0]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]], RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ")"]]]], "]"]]]], RowBox[List["4", " ", SqrtBox["2"]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]]]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["4", " ", SqrtBox["2"]]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "4"]]], ")"]]]], "]"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "4"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", FractionBox["1", "4"], ",", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "4"]]]]], "}"]]]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "16"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SqrtBox["\[Pi]"], SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["16", " "]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]]]]

 MathML Form

 K ν ( - 1 4 z ) ker ν ( z ) - 1 16 - 3 π ν 4 π ( - 1 4 z ) - ν ( + cot ( π ν ) ) z ν G 0 , 4 3 , 0 ( - z 4 64 0 , 1 2 , ν 2 , - ν 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["0", ",", "4"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] + 1 16 ( 3 π ν 4 π z - ν ( - 1 4 z ) ν csc ( π ν ) ) G 0 , 4 3 , 0 ( - z 4 64 0 , 1 2 , - ν 2 , ν 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["0", ",", "4"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - - 3 π ν 4 π 5 / 2 z - ν ( - 1 4 z ) ν csc ( π ν ) csc ( π ( ν + 1 4 ) ) 4 2 G 3 , 5 2 , 1 ( z 2 1 2 , 1 4 , - ν - 1 4 0 , - ν , ν , 1 4 , - ν - 1 4 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Nu]", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] + 3 π ν 4 π 5 / 2 z ν ( - 1 4 z ) - ν ( - + cot ( π ν ) ) csc ( π ( ν + 3 4 ) ) 4 2 G 3 , 5 2 , 1 ( z 2 1 2 , 1 4 , ν - 1 4 0 , ν , 1 4 , - ν , ν - 1 4 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Nu]", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] /; ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] - π 2 < arg ( z ) 0 Condition BesselK ν -1 1 4 z KelvinKer ν z -1 1 16 -1 3 ν 4 -1 1 2 -1 1 4 z -1 ν ν z ν MeijerG 0 1 2 ν 2 -1 -1 ν 2 -1 -1 z 4 64 -1 1 16 3 ν 4 -1 1 2 z -1 ν -1 1 4 z ν ν MeijerG 0 1 2 -1 ν 2 -1 ν 2 -1 -1 z 4 64 -1 -1 -1 3 ν 4 -1 5 2 z -1 ν -1 1 4 z ν ν ν 1 4 4 2 1 2 -1 MeijerG 1 2 1 4 -1 ν -1 1 4 0 -1 ν ν 1 4 -1 ν -1 1 4 z 2 3 ν 4 -1 5 2 z ν -1 1 4 z -1 ν -1 ν ν 3 4 4 2 1 2 -1 MeijerG 1 2 1 4 ν -1 1 4 0 ν 1 4 -1 ν ν -1 1 4 z 2 ν Inequality -1 2 -1 z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z_"]]]], "]"]], " ", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List["4", " ", SqrtBox["2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "4"]]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", FractionBox["1", "4"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", 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RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]], "&&", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]]]]]]

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

 2007-05-02