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variants of this functions
EllipticE




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/08.04.06.0071.01









  


  








Input Form





EllipticE[z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Sqrt[(-m) Sin[z]^2]/Sin[z]) (1 - Cos[z] + (-1 + 2 Log[Cos[z/2]^2] - Log[-4 m Sin[z]^2])/(4 m) + (1/(64 m^2)) (3 + 4 Cos[z] Csc[z]^2 + 4 Log[Cos[z/2]^2] - 2 Log[-4 m Sin[z]^2]) + (6 + 2 Cos[z] Csc[z]^2 (3 + 2 Csc[z]^2) + 6 Log[Cos[z/2]^2] - 3 Log[-4 m Sin[z]^2])/(256 m^3) + \[Ellipsis]) + 2 Round[Re[z]/Pi] (Sqrt[-m] + (Log[-m]/(4 Sqrt[-m])) (1 + 1/(8 m) + 3/(64 m^2) + \[Ellipsis]) + (1/Sqrt[-m]) (1/4 + Log[2] + (-3 + 8 Log[2])/(64 m) + (-3 + 6 Log[2])/(128 m^2) + \[Ellipsis])) /; (Abs[m] -> Infinity)








Standard Form





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MathML Form







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</annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </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> FractionBox </ci> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> m </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> cos </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> cos </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 4 </ms> </list> </apply> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> m </ms> </list> </apply> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> RowBox </ci> <list> <ms> cos </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> cos </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> ) </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> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 4 </ms> </list> </apply> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <ms> 3 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 64 </ms> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> cos </ms> <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> 2 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <ms> 3 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 6 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> cos </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 4 </ms> </list> </apply> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <ms> 6 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 256 </ms> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 3 </ms> </apply> </list> </apply> </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> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> m </ms> </list> </apply> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> m </ms> </list> </apply> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> 8 </ms> <ms> m </ms> </list> </apply> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <apply> <ci> RowBox </ci> <list> <ms> 64 </ms> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <ms> + </ms> <ms> &#8230; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> m </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 8 </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> <apply> <ci> RowBox </ci> <list> <ms> 64 </ms> <ms> m </ms> </list> </apply> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 6 </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> <apply> <ci> RowBox </ci> <list> <ms> 128 </ms> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> </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> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#62979; </ms> <ms> m </ms> <ms> &#62980; </ms> </list> </apply> <ms> &#62754; </ms> <ms> &#8734; </ms> </list> </apply> <ms> ) </ms> </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[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Cos", "[", "z", "]"]], "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "2"], "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], RowBox[List["4", " ", "m"]]], "+", FractionBox[RowBox[List["3", "+", RowBox[List["4", " ", RowBox[List["Cos", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "2"], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]]]], RowBox[List["64", " ", SuperscriptBox["m", "2"]]]], "+", FractionBox[RowBox[List["6", "+", RowBox[List["2", " ", RowBox[List["Cos", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"]]]]], ")"]]]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "2"], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]]]], RowBox[List["256", " ", SuperscriptBox["m", "3"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["Sin", "[", "z", "]"]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", "m"]]], "+", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["1", RowBox[List["8", " ", "m"]]], "+", FractionBox["3", RowBox[List["64", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["-", "m"]]]]]], "+", FractionBox[RowBox[List[FractionBox["1", "4"], "+", RowBox[List["Log", "[", "2", "]"]], "+", FractionBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["8", " ", RowBox[List["Log", "[", "2", "]"]]]]]], RowBox[List["64", " ", "m"]]], "+", FractionBox[RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", "2", "]"]]]]]], RowBox[List["128", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], SqrtBox[RowBox[List["-", "m"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]


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





2007-05-02