Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











JacobiCN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiCN[z,m] > Series representations > Generalized power series > Expansions at m==0





http://functions.wolfram.com/09.26.06.0012.01









  


  










Input Form





JacobiCN[z, m] \[Proportional] Cos[z] + (1/8) Sin[z] (2 z - Sin[2 z]) m + (1/256) ((-(9 + 8 z^2)) Cos[z] + 8 Cos[3 z] + Cos[5 z] + 16 z Sin[z] + 12 z Sin[3 z]) m^2 + \[Ellipsis] /; (m -> 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "8"], RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "z"]], "-", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], "m"]], "+", RowBox[List[FractionBox["1", "256"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["9", "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["8", " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", RowBox[List["Cos", "[", RowBox[List["5", " ", "z"]], "]"]], "+", RowBox[List["16", " ", "z", " ", RowBox[List["Sin", "[", "z", "]"]]]], "+", RowBox[List["12", " ", "z", " ", RowBox[List["Sin", "[", RowBox[List["3", " ", "z"]], "]"]]]]]], ")"]], SuperscriptBox["m", "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", "0"]], ")"]]]]]]










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> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 256 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <cos /> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <sin /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 256 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <cos /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <ci> z </ci> <apply> <sin /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> <apply> <sin /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "z"]], "-", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], " ", "m"]], "+", RowBox[List[FractionBox["1", "256"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["9", "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], " ", RowBox[List["Cos", "[", "z", "]"]]]], "+", RowBox[List["8", " ", RowBox[List["Cos", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", RowBox[List["Cos", "[", RowBox[List["5", " ", "z"]], "]"]], "+", RowBox[List["16", " ", "z", " ", RowBox[List["Sin", "[", "z", "]"]]]], "+", RowBox[List["12", " ", "z", " ", RowBox[List["Sin", "[", RowBox[List["3", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox["m", "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", "0"]], ")"]]]]]]]]










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





2007-05-02