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http://functions.wolfram.com/08.03.06.0018.01
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EllipticPi[n, m] \[Proportional] EllipticK[m] -
((EllipticE[m] - EllipticK[m])/m) n -
((2 (1 + m) EllipticE[m] - (2 + m) EllipticK[m])/(3 m^2)) n^2 + O[n^3]
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Cell[BoxData[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], "-", RowBox[List["EllipticK", "[", "m", "]"]]]], "m"], "n"]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], RowBox[List["3", " ", SuperscriptBox["m", "2"]]]], SuperscriptBox["n", "2"]]], "+", RowBox[List["O", "[", SuperscriptBox["n", "3"], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> m </mi> </mfrac> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> n </mi> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], "-", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]], " ", "n"]], "m"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]], " ", SuperscriptBox["n", "2"]]], RowBox[List["3", " ", SuperscriptBox["m", "2"]]]], "+", SuperscriptBox[RowBox[List["O", "[", "n", "]"]], "3"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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