|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.13.13.0009.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Derivative[2][w][\[Nu]] - ((2 Derivative[1][h][\[Nu]])/h[\[Nu]] -
(2 Derivative[1][g][\[Nu]])/g[\[Nu]] + Derivative[2][g][\[Nu]]/
Derivative[1][g][\[Nu]]) Derivative[1][w][\[Nu]] +
(ArcCos[z]^2 Derivative[1][g][\[Nu]]^2 -
(2 Derivative[1][g][\[Nu]] Derivative[1][h][\[Nu]])/
(g[\[Nu]] h[\[Nu]]) + (2 Derivative[1][h][\[Nu]]^2)/h[\[Nu]]^2 +
(Derivative[1][h][\[Nu]] Derivative[2][g][\[Nu]])/
(h[\[Nu]] Derivative[1][g][\[Nu]]) - Derivative[2][h][\[Nu]]/h[\[Nu]])
w[\[Nu]] == 0 /; w[\[Nu]] ==
Subscript[c, 1] h[\[Nu]] GegenbauerC[g[\[Nu]], z] +
Subscript[c, 2] (h[\[Nu]]/g[\[Nu]]) ChebyshevU[g[\[Nu]], z]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]], "-", 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["2", " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]]]], RowBox[List["g", "[", "\[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["2", " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]]]], RowBox[List[RowBox[List["g", "[", "\[Nu]", "]"]], " ", RowBox[List["h", "[", "\[Nu]", "]"]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]], "2"]]], SuperscriptBox[RowBox[List["h", "[", "\[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[SuperscriptBox["h", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]], RowBox[List["h", "[", "\[Nu]", "]"]]]]], ")"]], RowBox[List["w", "[", "\[Nu]", "]"]]]]]], "\[Equal]", "0"]], " ", "/;", " ", RowBox[List[RowBox[List["w", "[", "\[Nu]", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["h", "[", "\[Nu]", "]"]], RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["g", "[", "\[Nu]", "]"]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], FractionBox[RowBox[List["h", "[", "\[Nu]", "]"]], RowBox[List["g", "[", "\[Nu]", "]"]]], RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["g", "[", "\[Nu]", "]"]], ",", "z"]], "]"]]]]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> w </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> h </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> g </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <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> ⁢ </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> h </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> h </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <msup> <mi> h </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> g </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> h </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mrow> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mfrac> <mrow> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </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> ν </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> g </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <ci> g </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> h </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <ci> h </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> ν </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> g </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> g </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> w </ci> <ci> ν </ci> </apply> </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> ν </ci> </bvar> <apply> <ci> g </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> h </ci> <ci> ν </ci> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> g </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> g </ci> <ci> ν </ci> </apply> <apply> <ci> h </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> h </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> h </ci> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> h </ci> <ci> ν </ci> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> ν </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> g </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> h </ci> <ci> ν </ci> </apply> <apply> <partialdiff /> <bvar> <ci> ν </ci> </bvar> <apply> <ci> g </ci> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> ν </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> h </ci> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <ci> h </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> ν </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> h </ci> <ci> ν </ci> </apply> <apply> <ci> GegenbauerC </ci> <apply> <ci> g </ci> <ci> ν </ci> </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> <times /> <apply> <ci> h </ci> <ci> ν </ci> </apply> <apply> <power /> <apply> <ci> g </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ChebyshevU </ci> <apply> <ci> g </ci> <ci> ν </ci> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| 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[FractionBox[RowBox[List["2", " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], RowBox[List["h", "[", "\[Nu]_", "]"]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], RowBox[List["g", "[", "\[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["2", " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], RowBox[List[RowBox[List["g", "[", "\[Nu]_", "]"]], " ", RowBox[List["h", "[", "\[Nu]_", "]"]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], "2"]]], SuperscriptBox[RowBox[List["h", "[", "\[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[SuperscriptBox["h", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], RowBox[List["h", "[", "\[Nu]_", "]"]]]]], ")"]], " ", 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["GegenbauerC", "[", RowBox[List[RowBox[List["g", "[", "\[Nu]", "]"]], ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SubscriptBox["c", "2"], " ", RowBox[List["h", "[", "\[Nu]", "]"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["g", "[", "\[Nu]", "]"]], ",", "z"]], "]"]]]], RowBox[List["g", "[", "\[Nu]", "]"]]]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|