Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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 generic point nu==nu0 > For the function itself





http://functions.wolfram.com/07.05.06.0035.01









  


  










Input Form





ChebyshevU[\[Nu], z] \[Proportional] (1/Sqrt[1 - z^2]) (Sin[ArcCos[z] (1 + Subscript[\[Nu], 0])] + ArcCos[z] Cos[ArcCos[z] (1 + Subscript[\[Nu], 0])] (\[Nu] - Subscript[\[Nu], 0]) - (1/2) ArcCos[z]^2 Sin[ArcCos[z] (1 + Subscript[\[Nu], 0])] (\[Nu] - Subscript[\[Nu], 0])^ 2) + O[(\[Nu] - Subscript[\[Nu], 0])^3]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "]"]], "+", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["ArcCos", "[", "z", "]"]], "2"], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "2"]]]]], ")"]]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "3"], "]"]]]]]]]]










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> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </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> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sin /> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <cos /> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <arccos /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <sin /> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </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[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], "]"]], "+", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["ArcCos", "[", "z", "]"]], "2"], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "2"]]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "3"]]]]]]]










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





2007-05-02