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] > Identities > Recurrence identities > Distant neighbors > Increasing





http://functions.wolfram.com/03.20.17.0007.01









  


  










Input Form





KelvinKer[\[Nu], z] == -((1/z^5) (Sqrt[2] (3 + \[Nu]) (-3 z^4 + 16 z^2 (8 + 6 \[Nu] + \[Nu]^2) + 16 (40 + 78 \[Nu] + 49 \[Nu]^2 + 12 \[Nu]^3 + \[Nu]^4)) KelvinKei[5 + \[Nu], z])) - (12 (2 + \[Nu]) (3 + \[Nu]) KelvinKei[6 + \[Nu], z])/z^2 + (1/z^5) (Sqrt[2] (3 + \[Nu]) (-3 z^4 - 16 z^2 (8 + 6 \[Nu] + \[Nu]^2) + 16 (40 + 78 \[Nu] + 49 \[Nu]^2 + 12 \[Nu]^3 + \[Nu]^4)) KelvinKer[5 + \[Nu], z]) - ((z^4 - 16 (24 + 50 \[Nu] + 35 \[Nu]^2 + 10 \[Nu]^3 + \[Nu]^4)) KelvinKer[6 + \[Nu], z])/z^4










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", SuperscriptBox["z", "5"]], RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["6", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "+", RowBox[List["16", " ", RowBox[List["(", RowBox[List["40", "+", RowBox[List["78", " ", "\[Nu]"]], "+", RowBox[List["49", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["12", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["5", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]]]], "-", FractionBox[RowBox[List["12", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["6", "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "2"]], "+", RowBox[List[FractionBox["1", SuperscriptBox["z", "5"]], RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["6", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "+", RowBox[List["16", " ", RowBox[List["(", RowBox[List["40", "+", RowBox[List["78", " ", "\[Nu]"]], "+", RowBox[List["49", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["12", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["5", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "4"], "-", RowBox[List["16", " ", RowBox[List["(", RowBox[List["24", "+", RowBox[List["50", " ", "\[Nu]"]], "+", RowBox[List["35", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["10", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["6", "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "4"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> ker </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 49 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 78 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 40 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> kei </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 5 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 49 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 78 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 40 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> ker </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 5 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> kei </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 6 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 35 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 50 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 24 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> ker </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 6 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> KelvinKer </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -3 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 49 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 78 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 40 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> KelvinKei </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 5 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -3 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 49 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 78 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 40 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> KelvinKer </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 5 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> KelvinKei </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 6 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 35 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 50 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 24 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> KelvinKer </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 6 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["6", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "+", RowBox[List["16", " ", RowBox[List["(", RowBox[List["40", "+", RowBox[List["78", " ", "\[Nu]"]], "+", RowBox[List["49", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["12", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["5", "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "5"]]]], "-", FractionBox[RowBox[List["12", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["6", "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["6", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "+", RowBox[List["16", " ", RowBox[List["(", RowBox[List["40", "+", RowBox[List["78", " ", "\[Nu]"]], "+", RowBox[List["49", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["12", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["5", "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "5"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "4"], "-", RowBox[List["16", " ", RowBox[List["(", RowBox[List["24", "+", RowBox[List["50", " ", "\[Nu]"]], "+", RowBox[List["35", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["10", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["6", "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "4"]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.