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
 











JacobiSN




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSN[z,m] > Series representations > Generalized power series > Expansions at m==1





http://functions.wolfram.com/09.36.06.0015.01









  


  








Input Form





JacobiSN[z, m] \[Proportional] Tanh[z] + (1/4) (z Sech[z]^2 - Tanh[z]) (m - 1) - (1/512) Sech[z]^3 (72 z Cosh[z] + 4 (-5 + 8 z^2) Sinh[z] - 19 Sinh[3 z] + Sinh[5 z]) (m - 1)^2 + \[Ellipsis] /; (m -> 1)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Tanh", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"]]], "-", RowBox[List["Tanh", "[", "z", "]"]]]], ")"]], RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox["1", "512"], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["72", " ", "z", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", "z", "]"]]]], "-", RowBox[List["19", " ", RowBox[List["Sinh", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", RowBox[List["Sinh", "[", RowBox[List["5", " ", "z"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]]]]]]








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> sn </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> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 512 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 72 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 19 </mn> <mo> &#8290; </mo> <mrow> <mi> sinh </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> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <msup> <mi> sech </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <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> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <tanh /> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <tanh /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 512 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 72 </cn> <ci> z </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </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'> -5 </cn> </apply> <apply> <sinh /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19 </cn> <apply> <sinh /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Tanh", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"]]], "-", RowBox[List["Tanh", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox["1", "512"], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["72", " ", "z", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", "z", "]"]]]], "-", RowBox[List["19", " ", RowBox[List["Sinh", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", RowBox[List["Sinh", "[", RowBox[List["5", " ", "z"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]]]]]]]]


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





2007-05-02