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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==0





http://functions.wolfram.com/08.04.06.0027.01









  


  








Input Form





EllipticE[z, m] \[Proportional] SeriesData[$CellContext`z, 0, {1, 0, Rational[1, 6] $CellContext`m, 0, Rational[1, 120] (-4 $CellContext`m + 9 $CellContext`m^2), 0, Rational[1, 5040] (16 $CellContext`m - 180 $CellContext`m^2 + 225 $CellContext`m^3), 0, Rational[1, 362880] (-64 $CellContext`m + 3024 $CellContext`m^2 - 12600 $CellContext`m^3 + 11025 $CellContext`m^4)}, 1, 11, 1] /; (z -> 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", InterpretationBox[RowBox[List["z", "-", FractionBox[RowBox[List["m", " ", SuperscriptBox["z", "3"]]], "6"], "-", RowBox[List[FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m"]], "-", "4"]], ")"]]]], "120"], " ", SuperscriptBox["z", "5"]]], "-", RowBox[List[FractionBox[RowBox[List["m", RowBox[List["(", RowBox[List["16", "-", RowBox[List["60", " ", "m"]], "+", RowBox[List["45", " ", SuperscriptBox["m", "2"]]]]], ")"]]]], "5040"], SuperscriptBox["z", "7"]]], "-", RowBox[List[FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "64"]], "+", RowBox[List["1008", " ", "m"]], "-", RowBox[List["2520", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["1575", " ", SuperscriptBox["m", "3"]]]]], ")"]]]], "362880"], SuperscriptBox["z", "9"]]], "+", "\[Ellipsis]"]], SeriesData[$CellContext`z, 0, List[1, 0, Times[Rational[1, 6], $CellContext`m], 0, Times[Rational[1, 120], Plus[Times[-4, $CellContext`m], Times[9, Power[$CellContext`m, 2]]]], 0, Times[Rational[1, 5040], Plus[Times[16, $CellContext`m], Times[-1, 180, Power[$CellContext`m, 2]], Times[225, Power[$CellContext`m, 3]]]], 0, Times[Rational[1, 362880], Plus[Times[-64, $CellContext`m], Times[3024, Power[$CellContext`m, 2]], Times[-1, 12600, Power[$CellContext`m, 3]], Times[11025, Power[$CellContext`m, 4]]]]], 1, 11, 1]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]








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> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mn> 6 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 120 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 16 </mn> <mo> - </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 5040 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 64 </mn> </mrow> <mo> + </mo> <mrow> <mn> 1008 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 2520 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1575 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 362880 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </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 /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <power /> <cn type='integer'> 120 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> m </ci> <apply> <plus /> <cn type='integer'> 16 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 60 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 5040 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> m </ci> <apply> <plus /> <cn type='integer'> -64 </cn> <apply> <times /> <cn type='integer'> 1008 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2520 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1575 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 362880 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </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[InterpretationBox[RowBox[List["$CellContext`z", "+", RowBox[List[FractionBox["1", "6"], " ", "$CellContext`m", " ", SuperscriptBox["$CellContext`z", "3"]]], "+", RowBox[List[FractionBox["1", "120"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "$CellContext`m"]], "+", RowBox[List["9", " ", SuperscriptBox["$CellContext`m", "2"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "5"]]], "+", RowBox[List[FractionBox["1", "5040"], " ", RowBox[List["(", RowBox[List[RowBox[List["16", " ", "$CellContext`m"]], "-", RowBox[List["180", " ", SuperscriptBox["$CellContext`m", "2"]]], "+", RowBox[List["225", " ", SuperscriptBox["$CellContext`m", "3"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "7"]]], "+", RowBox[List[FractionBox["1", "362880"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "64"]], " ", "$CellContext`m"]], "+", RowBox[List["3024", " ", SuperscriptBox["$CellContext`m", "2"]]], "-", RowBox[List["12600", " ", SuperscriptBox["$CellContext`m", "3"]]], "+", RowBox[List["11025", " ", SuperscriptBox["$CellContext`m", "4"]]]]], ")"]], " ", SuperscriptBox["$CellContext`z", "9"]]], "+", InterpretationBox[SuperscriptBox[RowBox[List["O", "[", "$CellContext`z", "]"]], "11"], SeriesData[$CellContext`z, 0, List[], 1, 11, 1], Rule[Editable, False]]]], SeriesData[$CellContext`z, 0, List[1, 0, Times[Rational[1, 6], $CellContext`m], 0, Times[Rational[1, 120], Plus[Times[-4, $CellContext`m], Times[9, Power[$CellContext`m, 2]]]], 0, Times[Rational[1, 5040], Plus[Times[16, $CellContext`m], Times[-1, 180, Power[$CellContext`m, 2]], Times[225, Power[$CellContext`m, 3]]]], 0, Times[Rational[1, 362880], Plus[Times[-64, $CellContext`m], Times[3024, Power[$CellContext`m, 2]], Times[-1, 12600, Power[$CellContext`m, 3]], Times[11025, Power[$CellContext`m, 4]]]]], 1, 11, 1], Rule[Editable, False]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]


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





2007-05-02