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
 











variants of this functions
EllipticE






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/08.04.06.0057.01









  


  










Input Form





EllipticE[z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Sin[z] + (1/2) (Sin[z] - ArcTanh[Sin[z]]) (m - 1) + (Sec[z]^2/32) (6 ArcTanh[Sin[z]] Cos[z]^2 - 3 Sin[z] - Sin[3 z]) (m - 1)^2 + \[Ellipsis]) + 2 Round[Re[z]/Pi] (1 + Log[1 - m] ((m - 1)/4) (1 + (3 (1 - m))/8 + (15/64) (1 - m)^2 + \[Ellipsis]) + ((m - 1)/4) (1 - 4 Log[2] + ((-13 + 24 Log[2])/16) (m - 1) - ((3 (-3 + 5 Log[2]))/16) (m - 1)^2 + \[Ellipsis])) /; (m -> 1) && !Element[(Pi + 2 Re[z])/(4 Pi), Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sec", "[", "z", "]"]], "2"], "32"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]], "-", RowBox[List["3", " ", RowBox[List["Sin", "[", "z", "]"]]]], "-", RowBox[List["Sin", "[", RowBox[List["3", " ", "z"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]], FractionBox[RowBox[List["m", "-", "1"]], "4"], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]], "8"], "+", RowBox[List[FractionBox["15", "64"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "4"], RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "13"]], "+", RowBox[List["24", " ", RowBox[List["Log", "[", "2", "]"]]]]]], "16"], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["3", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]]]], "16"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]], "]"]]]]]]]]










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> E </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> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8969; </mo> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </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> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 32 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sec </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </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> <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> <mtext> </mtext> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8969; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 8 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 15 </mn> <mn> 64 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 13 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <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> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <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> <mo> &#8743; </mo> <mrow> <mo> &#172; </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> &#960; </mi> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> E </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> &#10072; </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#8733; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#8970; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#960; </ms> </apply> <ms> &#8969; </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> tanh </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 32 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sec </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 6 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> tanh </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> cos </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> &#8230; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> &#8970; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#960; </ms> </apply> <ms> &#8969; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> 8 </ms> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 15 </ms> <ms> 64 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> &#8230; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 16 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 24 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <ms> 13 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 16 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 5 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <ms> 3 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> &#8230; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> &#62754; </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#172; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <ms> &#960; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> &#960; </ms> </list> </apply> </apply> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <list /> <integers /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "32"], " ", SuperscriptBox[RowBox[List["Sec", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]], "-", RowBox[List["3", " ", RowBox[List["Sin", "[", "z", "]"]]]], "-", RowBox[List["Sin", "[", RowBox[List["3", " ", "z"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]], "8"], "+", RowBox[List[FractionBox["15", "64"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "13"]], "+", RowBox[List["24", " ", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]], "&&", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]]]]










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





2007-05-02