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variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.13.13.0013.01









  


  










Input Form





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










Standard Form





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MathML Form







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</mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </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> s </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> r </mi> <mi> &#957; </mi> </msup> </mrow> <mrow> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> s </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> r </mi> <mi> &#957; </mi> </msup> </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> <ln /> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <ci> s </ci> </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 /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <arccos /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> r </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> s </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> r </ci> </apply> </apply> </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> s </ci> <ci> &#957; </ci> </apply> <apply> <ci> GegenbauerC </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> s </ci> <ci> &#957; </ci> </apply> <apply> <ci> ChebyshevU </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> &#957; </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["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "s_", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "\[Nu]_"]]], " ", SuperscriptBox[RowBox[List["ArcCos", "[", "z_", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "r_", "]"]], "2"]]], "+", RowBox[List[RowBox[List["Log", "[", "s_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r_", "]"]]]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "\[Nu]_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "\[Nu]", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["s", "\[Nu]"], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["r", "\[Nu]"]]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["r", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["s", "\[Nu]"], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["r", "\[Nu]"]]], ",", "z"]], "]"]]]]]]]]]]]]]]










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





2007-05-02





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