Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.06.13.0004.01









  


  










Input Form





Wronskian[Fibonacci[\[Nu], g[z]], (1/(4 + g[z]^2)^(1/4)) LegendreP[-(1/2) + \[Nu], 1/2, 2, (I g[z])/2], z] == (-(((3 + E^(2 I \[Nu] Pi)) \[Nu])/(E^((1/2) I \[Nu] Pi) (Sqrt[Pi] (4 + g[z]^2)^(3/2))))) Derivative[1][g][z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Wronskian", "[", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]], ",", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], RowBox[List["1", "/", "4"]]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", FractionBox["1", "2"], ",", "2", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["g", "[", "z", "]"]]]], "2"]]], "]"]]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List["3", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]]]]], ")"]], " ", "\[Nu]"]], RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "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> <msub> <mi> W </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mroot> <mrow> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> <mn> 4 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msubsup> <mo> ( </mo> <semantics> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; &quot;, RowBox[List[&quot;g&quot;, &quot;(&quot;, &quot;z&quot;, &quot;)&quot;]]]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> z </ci> </apply> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> g </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <ci> &#957; </ci> <pi /> </apply> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> &#957; </ci> <pi /> </apply> </apply> </apply> <ci> &#957; </ci> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> g </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Wronskian", "[", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", RowBox[List["g", "[", "z_", "]"]]]], "]"]], ",", FractionBox[RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]_"]], ",", FractionBox["1", "2"], ",", "2", ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["g", "[", "z_", "]"]]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z_", "]"]], "2"]]], ")"]], RowBox[List["1", "/", "4"]]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List["3", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]]]]], ")"]], " ", "\[Nu]"]], ")"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.