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
KelvinBer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinBer[nu,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > Special cases





http://functions.wolfram.com/03.18.06.0024.01









  


  










Input Form





KelvinBer[-2 n - 1, z] == (((-1)^Floor[(n + 1)/2] 2^(-(3/2) - 2 n) z^(1 + 2 n))/(2 n + 1)!) (1 - z^4/(64 (1 + n) (3 + 2 n)) + z^8/(24576 (1 + n) (2 + n) (3 + 2 n) (5 + 2 n)) + O[z^12]) + (((-1)^Floor[n/2] 2^(-(7/2) - 2 n) z^(3 + 2 n))/(2 n + 2)!) (1 - z^4/(192 (2 + n) (3 + 2 n)) + z^8/(122880 (2 + n) (3 + n) (3 + 2 n) (5 + 2 n)) + O[z^12]) /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "-", "1"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "+", "1"]], "2"], "]"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["64", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["24576", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", "12"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "2"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["192", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["122880", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", "12"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> ber </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 8 </mn> </msup> <mrow> <mn> 24576 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 12 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mrow> <mn> 192 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 8 </mn> </msup> <mrow> <mn> 122880 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 12 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> KelvinBer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 24576 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 12 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 192 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 122880 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 12 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], "-", "1"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "+", "1"]], "2"], "]"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["64", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["24576", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "12"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["z", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["192", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["122880", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "12"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.