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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 generic point m==m0 > For the function itself





http://functions.wolfram.com/08.04.06.0047.01









  


  










Input Form





EllipticE[z, m] \[Proportional] EllipticE[z, Subscript[m, 0]] + ((EllipticE[z, Subscript[m, 0]] - EllipticF[z, Subscript[m, 0]])/ (2 Subscript[m, 0])) (m - Subscript[m, 0]) + (1/(16 (-1 + Subscript[m, 0]) Subscript[m, 0]^2 Sqrt[1 - Subscript[m, 0] Sin[z]^2])) (2 Sqrt[1 - Subscript[m, 0] Sin[z]^2] ((-(-2 + Subscript[m, 0])) EllipticE[z, Subscript[m, 0]] + 2 (-1 + Subscript[m, 0]) EllipticF[z, Subscript[m, 0]]) - Subscript[m, 0] Sin[2 z]) (m - Subscript[m, 0])^2 + \[Ellipsis] /; (m -> Subscript[m, 0])










Standard Form





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MathML Form







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</mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </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> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticE </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticF </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </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["EllipticE", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]]]], RowBox[List["2", " ", SubscriptBox["mm", "0"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SubscriptBox["mm", "0"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SubscriptBox["mm", "0"]]], ")"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["mm", "0"]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]]]]]], ")"]]]], "-", RowBox[List[SubscriptBox["mm", "0"], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]], "2"]]], RowBox[List["16", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["mm", "0"]]], ")"]], " ", SubsuperscriptBox["mm", "0", "2"], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SubscriptBox["mm", "0"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", SubscriptBox["mm", "0"]]], ")"]]]]]]]]










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





2007-05-02





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