Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
KelvinKei






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKei[nu,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > General case





http://functions.wolfram.com/03.19.06.0020.01









  


  










Input Form





KelvinKei[\[Nu], z] \[Proportional] Piecewise[{{-Pi/4, \[Nu] == 0}, {(-(-1)^(Abs[\[Nu]]/4)) 2^(-3 + Abs[\[Nu]]) z^(2 - Abs[\[Nu]]) (-2 + Abs[\[Nu]])!, Element[\[Nu]/4, Integers]}, {((-1 + I) (-1)^(\[Nu] - 1) (-1)^(\[Nu]/4) 2^(-2 + Abs[\[Nu]]) (-1 + Abs[\[Nu]])!)/z^Abs[\[Nu]], Element[(\[Nu] - 1)/4, Integers]}, {((-1)^UnitStep[(\[Nu] - 2)/4] I (-1)^(\[Nu]/4) 2^(-1 + Abs[\[Nu]]) (-1 + Abs[\[Nu]])!)/z^Abs[\[Nu]], Element[(\[Nu] - 2)/4, Integers]}, {((1 + I) (-1)^(\[Nu] - 1) (-1)^(\[Nu]/4) 2^(-2 + Abs[\[Nu]]) (-1 + Abs[\[Nu]])!)/z^Abs[\[Nu]], Element[(\[Nu] - 3)/4, Integers]}}, (-2^(-1 - \[Nu])) z^\[Nu] Gamma[-\[Nu]] Sin[(Pi \[Nu])/4] - (2^(-1 + \[Nu]) Gamma[\[Nu]] Sin[(3 Pi \[Nu])/4])/z^\[Nu]] /; (z -> 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], "/", "4"]], ",", " ", RowBox[List["\[Nu]", "\[Equal]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "/", "4"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], ",", RowBox[List[FractionBox["\[Nu]", "4"], "\[Element]", "Integers"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[ImaginaryI]"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], ",", RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "1"]], "4"], "\[Element]", "Integers"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["UnitStep", "[", FractionBox[RowBox[List["\[Nu]", "-", "2"]], "4"], "]"]]], "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], ",", RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "2"]], "4"], "\[Element]", "Integers"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], ",", RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "3"]], "4"], "\[Element]", "Integers"]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "4"], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> kei </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> </mtd> <mtd> <mrow> <mi> &#957; </mi> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mn> 4 </mn> </mfrac> </msup> </mrow> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> </mrow> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> KelvinKei </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <piecewise> <piece> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </piece> <piece> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </piece> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> UnitStep </ci> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <imaginaryi /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </piece> <piece> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </piece> <otherwise> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> &#957; </ci> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], RowBox[List["\[Nu]", "\[Equal]", "0"]]], List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List["Abs", "[", "\[Nu]", "]"]], "4"]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], RowBox[List[FractionBox["\[Nu]", "4"], "\[Element]", "Integers"]]], List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "1"]], "4"], "\[Element]", "Integers"]]], List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["UnitStep", "[", FractionBox[RowBox[List["\[Nu]", "-", "2"]], "4"], "]"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "2"]], "4"], "\[Element]", "Integers"]]], List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], RowBox[List[FractionBox[RowBox[List["\[Nu]", "-", "3"]], "4"], "\[Element]", "Integers"]]], List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "4"], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.