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variants of this functions
ChebyshevU






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.05.13.0012.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[Nu]", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "r", "+", RowBox[List["2", " ", "s"]]]], ")"]], " ", "\[Nu]", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["s", " ", RowBox[List["(", RowBox[List["r", "+", "s"]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", "2"], " ", SuperscriptBox["\[Nu]", RowBox[List["2", " ", "r"]]], " ", SuperscriptBox[RowBox[List["ArcCos", "[", "z", "]"]], "2"]]]]], ")"]], " ", RowBox[List["w", "[", "\[Nu]", "]"]]]]]], "\[Equal]", "0"]], " ", "/;", " ", RowBox[List[RowBox[List["w", "[", "\[Nu]", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["\[Nu]", "s"], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["\[Nu]", "r"]]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["\[Nu]", "s"], " ", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["\[Nu]", "r"]]], ",", "z"]], "]"]]]]]]]]]]]]










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> <mrow> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mn> 2 </mn> </msup> <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> <msup> <mi> &#957; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mi> s </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mi> r </mi> </msup> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> T </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mi> r </mi> </msup> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> &#957; </ci> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <arccos /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <times /> <ci> s </ci> <apply> <plus /> <ci> r </ci> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> &#957; </ci> <ci> s </ci> </apply> <apply> <ci> ChebyshevU </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> &#957; </ci> <ci> r </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#957; </ci> <ci> s </ci> </apply> <apply> <ci> ChebyshevT </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> &#957; </ci> <ci> r </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2007-05-02





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