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; }

 KelvinKei

 http://functions.wolfram.com/03.19.26.0040.01

 Input Form

 BesselJ[\[Nu], (-1)^(1/4) z] KelvinKei[\[Nu], z] == ((1/8) E^((3/4) I Pi \[Nu]) Sqrt[Pi] ((-1)^(1/4) z)^\[Nu] ((-I) MeijerG[{{}, {(3 \[Nu] + 1)/2}}, {{0, 1/2, \[Nu]/2}, {-(\[Nu]/2), (3 \[Nu] + 1)/2}}, z^4/64] - MeijerG[{{}, {(3 \[Nu])/2}}, {{0, 1/2, \[Nu]/2}, {-(\[Nu]/2), (3 \[Nu])/2}}, z^4/64] - (1/(Pi Sqrt[2])) (MeijerG[{{1/4, 3/4}, {}}, {{1/2, \[Nu]/2, \[Nu]/2 + 1/2}, {-(\[Nu]/2), 1/2 - \[Nu]/2, 0}}, z^4/16] - I MeijerG[{{1/4, 3/4}, {}}, {{0, \[Nu]/2, (\[Nu] + 1)/2}, {1/2, -(\[Nu]/2), (1 - \[Nu])/2}}, z^4/16])))/z^\[Nu] /; -(Pi/4) <= Arg[z] <= Pi/4

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]], RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["3", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", 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["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox[RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], "2"]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]]]], "-", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]], "-", RowBox[List[FractionBox["1", RowBox[List["\[Pi]", SqrtBox["2"]]]], RowBox[List["(", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"], ",", RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", "0"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]], " ", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["\[Nu]", "2"], ",", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "4"]]]]]]]

 MathML Form

 J ν ( - 1 4 z ) kei ν ( z ) 1 8 3 π ν 4 π z - ν ( - 1 4 z ) ν ( - G 1 , 5 3 , 0 ( z 4 64 1 2 ( 3 ν + 1 ) 0 , 1 2 , ν 2 , - ν 2 , 1 2 ( 3 ν + 1 ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "5"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "4"], "64"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]], 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]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - G 1 , 5 3 , 0 ( z 4 64 3 ν 2 0 , 1 2 , ν 2 , - ν 2 , 3 ν 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "5"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "4"], "64"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]], 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]], ",", TagBox[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - 1 π 2 ( G 2 , 6 3 , 2 ( z 4 16 1 4 , 3 4 1 2 , ν 2 , ν 2 + 1 2 , - ν 2 , 1 2 - ν 2 , 0 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "6"]], RowBox[List["3", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "4"], "16"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["3", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[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"], "+", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - G 2 , 6 3 , 2 ( z 4 16 1 4 , 3 4 0 , ν 2 , ν + 1 2 , 1 2 , - ν 2 , 1 - ν 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "6"]], RowBox[List["3", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "4"], "16"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["3", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], 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[RowBox[List["1", "-", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] ) ) /; - π 4 arg ( z ) π 4 Condition BesselJ ν -1 1 4 z KelvinKei ν z 1 8 3 ν 4 -1 1 2 z -1 ν -1 1 4 z ν -1 MeijerG 1 2 3 ν 1 0 1 2 ν 2 -1 -1 ν 2 -1 1 2 3 ν 1 z 4 64 -1 -1 MeijerG 3 ν 2 -1 0 1 2 ν 2 -1 -1 ν 2 -1 3 ν 2 -1 z 4 64 -1 -1 1 2 1 2 -1 MeijerG 1 4 3 4 1 2 ν 2 -1 ν 2 -1 1 2 -1 ν 2 -1 1 2 -1 ν 2 -1 0 z 4 16 -1 -1 MeijerG 1 4 3 4 0 ν 2 -1 ν 1 2 -1 1 2 -1 ν 2 -1 1 -1 ν 2 -1 z 4 16 -1 -1 4 -1 z 4 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z_"]]]], "]"]], " ", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", 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["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], ")"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], ")"]]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]]]], "-", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Nu]", "2"], ",", RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", "0"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["\[Nu]", "2"], ",", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]]]]]], RowBox[List["\[Pi]", " ", SqrtBox["2"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "4"]]]]]]]]]

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

 2007-05-02

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