Inverse Weierstrass elliptic function: Series representations (formula 09.22.06.0001)

InverseWeierstrassP

$Mathematica Notation$

Elliptic Functions

InverseWeierstrassP[z,{g₂,g₃}]

Series representations

Generalized power series

Expansions at z==0

http://functions.wolfram.com/09.22.06.0001.01

Input Form

InverseWeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] == SeriesData[$CellContext`z, ComplexInfinity, {1, 0, 0, 0, Rational[1, 40] Subscript[$CellContext`g, 2], 0, Rational[1, 56] Subscript[$CellContext`g, 3], 0, Rational[1, 384] Subscript[$CellContext`g, 2]^2, 0, Rational[3, 704] Subscript[$CellContext`g, 2] Subscript[$CellContext`g, 3], 0, Rational[1, 13312] (5 Subscript[$CellContext`g, 2]^3 + 24 Subscript[$CellContext`g, 3]^2), 0, Rational[1, 1024] Subscript[$CellContext`g, 2]^2 Subscript[$CellContext`g, 3], 0, Rational[5, 557056] (7 Subscript[$CellContext`g, 2]^4 + 96 Subscript[$CellContext`g, 2] Subscript[$CellContext`g, 3]^2), 0, Rational[5, 155648] (7 Subscript[$CellContext`g, 2]^3 Subscript[$CellContext`g, 3] + 8 Subscript[$CellContext`g, 3]^3), 0, Rational[1, 262144] (3 Subscript[$CellContext`g, 2]^5 + 80 Subscript[$CellContext`g, 2]^2 Subscript[$CellContext`g, 3]^2)}, 1, 23, 2]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["InverseWeierstrassP", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", InterpretationBox[RowBox[List[SqrtBox[FractionBox["1", "z"]], "+", RowBox[List[FractionBox["1", "40"], " ", SubscriptBox["g", "2"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["5", "/", "2"]]]]], "+", RowBox[List[FractionBox["1", "56"], " ", SubscriptBox["g", "3"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["7", "/", "2"]]]]], "+", RowBox[List[FractionBox["1", "384"], " ", SubsuperscriptBox["g", "2", "2"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["9", "/", "2"]]]]], "+", RowBox[List[FractionBox["3", "704"], " ", SubscriptBox["g", "2"], " ", SubscriptBox["g", "3"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["11", "/", "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["5", " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["24", " ", SubsuperscriptBox["g", "3", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["13", "/", "2"]]]]], "13312"], "+", FractionBox[RowBox[List[SubsuperscriptBox["g", "2", "2"], " ", SubscriptBox["g", "3"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["15", "/", "2"]]]]], "1024"], "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", SubsuperscriptBox["g", "2", "4"]]], "+", RowBox[List["96", " ", SubscriptBox["g", "2"], " ", SubsuperscriptBox["g", "3", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["17", "/", "2"]]]]], "557056"], "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", SubsuperscriptBox["g", "2", "3"], " ", SubscriptBox["g", "3"]]], "+", RowBox[List["8", " ", SubsuperscriptBox["g", "3", "3"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["19", "/", "2"]]]]], "155648"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", SubsuperscriptBox["g", "2", "5"]]], "+", RowBox[List["80", " ", SubsuperscriptBox["g", "2", "2"], " ", SubsuperscriptBox["g", "3", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["21", "/", "2"]]]]], "262144"], "+", InterpretationBox[RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["23", "/", "2"]]], "]"]], SeriesData[$CellContext`z, DirectedInfinity[], List[], 1, 23, 2]]]], SeriesData[$CellContext`z, DirectedInfinity[], List[1, 0, 0, 0, Times[Rational[1, 40], Subscript[$CellContext`g, 2]], 0, Times[Rational[1, 56], Subscript[$CellContext`g, 3]], 0, Times[Rational[1, 384], Power[Subscript[$CellContext`g, 2], 2]], 0, Times[Rational[3, 704], Subscript[$CellContext`g, 2], Subscript[$CellContext`g, 3]], 0, Times[Rational[1, 13312], Plus[Times[5, Power[Subscript[$CellContext`g, 2], 3]], Times[24, Power[Subscript[$CellContext`g, 3], 2]]]], 0, Times[Rational[1, 1024], Power[Subscript[$CellContext`g, 2], 2], Subscript[$CellContext`g, 3]], 0, Times[Rational[5, 557056], Plus[Times[7, Power[Subscript[$CellContext`g, 2], 4]], Times[96, Subscript[$CellContext`g, 2], Power[Subscript[$CellContext`g, 3], 2]]]], 0, Times[Rational[5, 155648], Plus[Times[7, Power[Subscript[$CellContext`g, 2], 3], Subscript[$CellContext`g, 3]], Times[8, Power[Subscript[$CellContext`g, 3], 3]]]], 0, Times[Rational[1, 262144], Plus[Times[3, Power[Subscript[$CellContext`g, 2], 5]], Times[80, Power[Subscript[$CellContext`g, 2], 2], Power[Subscript[$CellContext`g, 3], 2]]]]], 1, 23, 2]]]]]]

MathML Form

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80 </mn> <mo> ⁢ </mo> <msubsup> <mi> g </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> <mo> ⁢ </mo> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> </mrow> <mn> 262144 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 21 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 23 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> SeriesData </ci> <ci> $CellContext`z </ci> <apply> <ci> DirectedInfinity </ci> </apply> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 40 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 56 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 384 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 3 <sep /> 704 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 13312 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 1024 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 5 <sep /> 557056 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 96 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 5 <sep /> 155648 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 262144 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 80 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </list> <cn type='integer'> 1 </cn> <cn type='integer'> 23 </cn> <cn type='integer'> 2 </cn> </apply> </annotation-xml> </semantics> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> InverseWeierstrassP </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <ci> SeriesData </ci> <ci> $CellContext`z </ci> <apply> <ci> DirectedInfinity </ci> </apply> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 40 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 56 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 384 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 3 <sep /> 704 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 13312 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 1024 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 5 <sep /> 557056 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 96 </cn> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 5 <sep /> 155648 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 262144 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 80 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> $CellContext`g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </list> <cn type='integer'> 1 </cn> <cn type='integer'> 23 </cn> <cn type='integer'> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

InverseWeierstrassP[{z₁,z₂},{g₂,g₃}]