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LucasL






Mathematica Notation

Traditional Notation









Integer Functions > LucasL[nu] > Differential equations > Ordinary linear differential equations and wronskians





http://functions.wolfram.com/04.22.13.0001.01









  


  










Input Form





Derivative[3][w][\[Nu]] + Log[GoldenRatio] Derivative[2][w][\[Nu]] + (Pi^2 - Log[GoldenRatio]^2) Derivative[1][w][\[Nu]] - Log[GoldenRatio] (Pi^2 + Log[GoldenRatio]^2) w[\[Nu]] == 0 /; w[\[Nu]] == Subscript[c, 1] LucasL[\[Nu]] + Subscript[c, 2] Fibonacci[\[Nu]] + (Subscript[c, 3] Sin[Pi \[Nu]])/ GoldenRatio^\[Nu]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]], "+", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "-", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "2"]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]", "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "2"]]], ")"]], " ", RowBox[List["w", "[", "\[Nu]", "]"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["w", "[", "\[Nu]", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["LucasL", "[", "\[Nu]", "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], RowBox[List["Fibonacci", "[", "\[Nu]", "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]]]]]]]]










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> w </mi> <semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;3&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <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> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </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> &#10869; </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> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mi> &#957; </mi> </msub> </mrow> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </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'> 3 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <ci> log </ci> <ci> GoldenRatio </ci> <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 /> <apply> <plus /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ln /> <ci> GoldenRatio </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> </bvar> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <ci> GoldenRatio </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ln /> <ci> GoldenRatio </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> w </ci> <ci> &#957; </ci> </apply> </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> <ci> LucasL </ci> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]], "+", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "-", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "2"]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "\[Nu]_", "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", "GoldenRatio", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", SuperscriptBox[RowBox[List["Log", "[", "GoldenRatio", "]"]], "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["LucasL", "[", "\[Nu]", "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", RowBox[List["Fibonacci", "[", "\[Nu]", "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]]]]]]]]]]










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





2007-05-02