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
KelvinKer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKer[nu,z] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations





http://functions.wolfram.com/03.20.16.0014.01









  


  










Input Form





KelvinKer[\[Nu], (z^4)^(1/4)] == ((1/8) z^(-2 - \[Nu]) (((-(2 + I^\[Nu] + E^((3 I \[Nu] Pi)/2))) z^2 + (-2 + I^\[Nu] + E^((3 I \[Nu] Pi)/2)) Sqrt[z^4]) (z^(2 \[Nu]) - (z^4)^(\[Nu]/2)) + 4 z^(2 \[Nu]) (z^2 + Sqrt[z^4] + (z^2 - Sqrt[z^4]) Cos[(3 \[Nu] Pi)/2])) KelvinKer[\[Nu], z])/(z^4)^(\[Nu]/4) + ((1/8) z^(-2 - \[Nu]) (-z^2 + Sqrt[z^4]) (I I^\[Nu] (-1 + (-1)^\[Nu]) (z^(2 \[Nu]) + (z^4)^(\[Nu]/2)) + 4 z^(2 \[Nu]) Sin[(3 \[Nu] Pi)/2]) KelvinKei[\[Nu], z])/ (z^4)^(\[Nu]/4) + ((1/32) z^(-2 - \[Nu]) (4 Pi ((-(-1 + E^((I \[Nu] Pi)/2) + E^((3 I \[Nu] Pi)/2))) z^(2 + 2 \[Nu]) + (1 + E^((I \[Nu] Pi)/2) + E^((3 I \[Nu] Pi)/2)) z^(2 \[Nu]) Sqrt[z^4] - z^2 (z^4)^(\[Nu]/2) - (z^4)^((1 + \[Nu])/2)) - 4 I I^\[Nu] (-1 + (-1)^\[Nu]) (-z^2 + Sqrt[z^4]) (z^(2 \[Nu]) - (z^4)^(\[Nu]/2)) Log[z] - 2 E^(I \[Nu] Pi) (-z^2 + Sqrt[z^4]) (z^(2 \[Nu]) - (z^4)^(\[Nu]/2)) Log[z^4] Sin[(\[Nu] Pi)/2]) KelvinBei[\[Nu], z])/(z^4)^(\[Nu]/4) + ((1/32) z^(-2 - \[Nu]) (-4 I I^\[Nu] (-1 + (-1)^\[Nu]) Pi z^(2 \[Nu]) (-z^2 + Sqrt[z^4]) - ((-(2 + I^\[Nu] + E^((3 I \[Nu] Pi)/2))) z^2 + (-2 + I^\[Nu] + E^((3 I \[Nu] Pi)/2)) Sqrt[z^4]) (z^(2 \[Nu]) + (z^4)^(\[Nu]/2)) (4 Log[z] - Log[z^4])) KelvinBer[\[Nu], z])/(z^4)^(\[Nu]/4) /; Element[\[Nu], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["1", "/", "4"]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "+", SqrtBox[SuperscriptBox["z", "4"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "+", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]], RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "32"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "4"]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]]]], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", SqrtBox[SuperscriptBox["z", "4"]]]], "-", RowBox[List[SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]]], ")"]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]], " ", RowBox[List["Log", "[", SuperscriptBox["z", "4"], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]], " ", RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], " ", "+", RowBox[List[FractionBox["1", "32"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"]]], ")"]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "+", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox["z", "4"], "]"]]]], ")"]]]]]], ")"]], RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]










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> ker </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mroot> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8520; </mi> <mi> &#957; </mi> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <msup> <mi> &#8520; </mi> <mi> &#957; </mi> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> ker </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 32 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8520; </mi> <mi> &#957; </mi> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <msup> <mi> &#8520; </mi> <mi> &#957; </mi> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> ber </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 32 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> bei </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </msqrt> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> kei </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> KelvinKer </ci> <ci> &#957; </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <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> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> KelvinKer </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 32 </cn> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <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> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <imaginaryi /> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <pi /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> -2 </cn> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> KelvinBer </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 32 </cn> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <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> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> &#957; </ci> <pi /> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> KelvinBei </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <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> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> KelvinKei </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> &#957; </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", SuperscriptBox[RowBox[List["(", SuperscriptBox["z_", "4"], ")"]], RowBox[List["1", "/", "4"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["-", FractionBox["\[Nu]", "4"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "+", SqrtBox[SuperscriptBox["z", "4"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["-", FractionBox["\[Nu]", "4"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "+", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]], " ", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "32"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["-", FractionBox["\[Nu]", "4"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]]]], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", SqrtBox[SuperscriptBox["z", "4"]]]], "-", RowBox[List[SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]]], ")"]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "-", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]], " ", RowBox[List["Log", "[", SuperscriptBox["z", "4"], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]], " ", RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "32"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["-", FractionBox["\[Nu]", "4"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"]]], ")"]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SqrtBox[SuperscriptBox["z", "4"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], "+", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "2"]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], "+", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "4"], ")"]], RowBox[List["\[Nu]", "/", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox["z", "4"], "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.