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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 z==infinity





http://functions.wolfram.com/08.04.06.0044.01









  


  








Input Form





EllipticE[z, m] \[Proportional] 2 EllipticE[m] Round[Re[z]/Pi] + (-1)^Round[Re[z]/Pi] (((Csc[z] Sqrt[-Sin[z]^2])/Sqrt[(-m) Sin[z]^2]) (1 - ((1 + m)/(6 m)) Csc[z]^2 - ((3 + 2 m + 3 m^2)/(120 m^2)) Csc[z]^4 + O[Csc[z]^6]) - Csc[z] Sqrt[-Sin[z]^2] Sqrt[(-m) Sin[z]^2] (1 - ((1 + m)/(2 m)) Csc[z]^2 - (1/8) (1 - 1/m)^2 Csc[z]^4 + O[Csc[z]^6]) + (1/2) Sqrt[-m] Csc[z] Sqrt[-Sin[z]^2] ((((I Sqrt[-Sin[z]^2])/Sqrt[Sin[z]^2]) (Sqrt[1/m] - 1/Sqrt[m]) - Sqrt[1/m] - 1/Sqrt[m]) EllipticE[m] + 2 EllipticE[1/m] + 2 ((1 - m)/m) EllipticK[1/m] + 2 I (1 - Sqrt[m/(m - 1)] Sqrt[(m - 1)/m]) (-EllipticE[1 - 1/m] + (1/m) EllipticK[1 - 1/m]))) /; (Abs[z] -> Infinity)








Standard Form





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MathML Form







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</ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> m </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </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> m </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> K </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> m </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> csc </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </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> </apply> <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> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </ms> </list> </apply> </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> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 8 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> m </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 4 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> O </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 6 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> csc </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </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> </apply> </list> </apply> <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> <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> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 6 </ms> <ms> m </ms> </list> </apply> </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> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </ms> </list> </apply> <ms> + </ms> <ms> 3 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 120 </ms> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 4 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> O </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> csc </ms> <ms> 6 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </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> E </ms> <ms> ( </ms> <ms> m </ms> <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> z </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[RowBox[List["2", " ", RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Csc", "[", "z", "]"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"]]], RowBox[List["6", " ", "m"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "4"]]], RowBox[List["120", " ", SuperscriptBox["m", "2"]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["Csc", "[", "z", "]"]], "]"]], "6"]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "-", RowBox[List[RowBox[List["Csc", "[", "z", "]"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"]]], RowBox[List["2", " ", "m"]]], "-", RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["1", "m"]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "4"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["Csc", "[", "z", "]"]], "]"]], "6"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List["-", "m"]]], " ", RowBox[List["Csc", "[", "z", "]"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SqrtBox[FractionBox["1", "m"]], "-", FractionBox["1", SqrtBox["m"]]]], ")"]]]], SqrtBox[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "-", SqrtBox[FractionBox["1", "m"]], "-", FractionBox["1", SqrtBox["m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["2", " ", RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]], "m"], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["m", RowBox[List["m", "-", "1"]]]], " ", SqrtBox[FractionBox[RowBox[List["m", "-", "1"]], "m"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]]]], "+", FractionBox[RowBox[List["EllipticK", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]], "m"]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]


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





2007-05-02