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
ChebyshevU






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ChebyshevU[nu,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Expansions in 1/z





http://functions.wolfram.com/07.05.06.0071.01









  


  










Input Form





ChebyshevU[\[Nu], z] \[Proportional] 2^\[Nu] z^\[Nu] (1 + (1 - \[Nu])/(4 z^2) + ((2 - \[Nu]) (3 - \[Nu]))/ (32 z^4) + O[1/z^6]) - 2^(-\[Nu] - 2) z^(-\[Nu] - 2) (1 + (3 + \[Nu])/(4 z^2) + ((4 + \[Nu]) (5 + \[Nu]))/(32 z^4) + O[1/z^6])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "6"]], "]"]]]], ")"]]]], " ", "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "2"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["3", "+", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "6"]], "]"]]]], ")"]]]]]]]]]]










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> U </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <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> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> ChebyshevU </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <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'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 3 </cn> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 4 </cn> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> 5 </cn> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "6"]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["3", "+", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "6"]], "]"]]]], ")"]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.