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variants of this functions
ChebyshevT






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ChebyshevT[nu,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself > With respect to nu





http://functions.wolfram.com/07.04.13.0010.01









  


  










Input Form





Derivative[2][w][\[Nu]] + (-((2 Derivative[1][h][\[Nu]])/h[\[Nu]]) - Derivative[2][g][\[Nu]]/Derivative[1][g][\[Nu]]) Derivative[1][w][\[Nu]] + (ArcCos[z]^2 Derivative[1][g][\[Nu]]^2 + (Derivative[1][h][\[Nu]] Derivative[2][g][\[Nu]])/ (h[\[Nu]] Derivative[1][g][\[Nu]]) + (2 Derivative[1][h][\[Nu]]^2 - h[\[Nu]] Derivative[2][h][\[Nu]])/ h[\[Nu]]^2) w[\[Nu]] == 0 /; w[\[Nu]] == Subscript[c, 1] h[\[Nu]] ChebyshevT[g[\[Nu]], z] + Subscript[c, 2] h[\[Nu]] ChebyshevU[g[\[Nu]], z]










Standard Form





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MathML Form







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</mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> g </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> g </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <arccos /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> g </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> g </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> g </ci> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <ci> h </ci> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> &#957; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], RowBox[List["h", "[", "\[Nu]_", "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcCos", "[", "z_", "]"]], "2"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], "2"]]], "+", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], RowBox[List[RowBox[List["h", "[", "\[Nu]_", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], "2"]]], "-", RowBox[List[RowBox[List["h", "[", "\[Nu]_", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]]]], SuperscriptBox[RowBox[List["h", "[", "\[Nu]_", "]"]], "2"]]]], ")"]], " ", RowBox[List["w", "[", "\[Nu]_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "\[Nu]", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", RowBox[List["h", "[", "\[Nu]", "]"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["g", "[", "\[Nu]", "]"]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", RowBox[List["h", "[", "\[Nu]", "]"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["g", "[", "\[Nu]", "]"]], ",", "z"]], "]"]]]]]]]]]]]]]]










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





2007-05-02