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variants of this functions
ChebyshevT




Mathematica Notation

Traditional Notation









Hypergeometric Functions > ChebyshevT[nu,z] > Series representations > Generalized power series > Expansions at generic point nu==nu0 > For the function itself





http://functions.wolfram.com/07.04.06.0032.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", SubscriptBox["\[Nu]", "0"]]], "]"]], "-", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", SubscriptBox["\[Nu]", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["ArcCos", "[", "z", "]"]], "2"], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", 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> T </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <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> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> </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> 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> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> </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> 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> <msub> <mi> &#957; </mi> <mn> 0 </mn> </msub> </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> <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> ChebyshevT </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <cos /> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <sin /> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </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> <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> <cos /> <apply> <times /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 0 </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> <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["ChebyshevT", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "-", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", 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["Cos", "[", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "3"]]]]]]]


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





2007-05-02