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http://functions.wolfram.com/08.03.06.0031.01
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EllipticPi[n, m] \[Proportional] EllipticPi[n, Subscript[m, 0]] +
(1/(2 (n - Subscript[m, 0]))) (EllipticE[Subscript[m, 0]]/
(Subscript[m, 0] - 1) + EllipticPi[n, Subscript[m, 0]])
(m - Subscript[m, 0]) + (1/2) ((3/(4 (-n + Subscript[m, 0])^2))
EllipticPi[n, Subscript[m, 0]] +
(1/(4 (-1 + Subscript[m, 0]) Subscript[m, 0] (-n + Subscript[m, 0])))
EllipticK[Subscript[m, 0]] +
((-n - (2 + n) Subscript[m, 0] + 4 Subscript[m, 0]^2)/
(4 (-1 + Subscript[m, 0])^2 Subscript[m, 0] (-n + Subscript[m, 0])^2))
EllipticE[Subscript[m, 0]]) (m - Subscript[m, 0])^2 + \[Ellipsis] /;
(m -> Subscript[m, 0])
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["m", "0"]]], "]"]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", SubscriptBox["m", "0"]]], ")"]]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["EllipticE", "[", SubscriptBox["m", "0"], "]"]], RowBox[List[SubscriptBox["m", "0"], "-", "1"]]], "+", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["m", "0"]]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]]]], " ", "+", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["3", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["m", "0"]]], ")"]], "2"]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["m", "0"]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["m", "0"]]], ")"]], " ", SubscriptBox["m", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["m", "0"]]], ")"]]]]], RowBox[List["EllipticK", "[", SubscriptBox["m", "0"], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", SubscriptBox["m", "0"]]], "+", RowBox[List["4", " ", SubsuperscriptBox["m", "0", "2"]]]]], ")"]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["m", "0"]]], ")"]], "2"], " ", SubscriptBox["m", "0"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["m", "0"]]], ")"]], "2"]]]], RowBox[List["EllipticE", "[", SubscriptBox["m", "0"], "]"]]]]]], " ", ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", SubscriptBox["m", "0"]]], ")"]]]]]]
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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> <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> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msubsup> <mi> m </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> - </mo> <mi> n </mi> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 3 </mn> <mtext> </mtext> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </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> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </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> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> EllipticPi </ci> <ci> n </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> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticE </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </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 /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticE </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </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> … </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>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["mm", "0"]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["EllipticE", "[", SubscriptBox["mm", "0"], "]"]], RowBox[List[SubscriptBox["mm", "0"], "-", "1"]]], "+", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["mm", "0"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", SubscriptBox["mm", "0"]]], ")"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["3", " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["mm", "0"]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["mm", "0"]]], ")"]], "2"]]]], "+", FractionBox[RowBox[List["EllipticK", "[", SubscriptBox["mm", "0"], "]"]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["mm", "0"]]], ")"]], " ", SubscriptBox["mm", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["mm", "0"]]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", SubscriptBox["mm", "0"]]], "+", RowBox[List["4", " ", SubsuperscriptBox["mm", "0", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", SubscriptBox["mm", "0"], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["mm", "0"]]], ")"]], "2"], " ", SubscriptBox["mm", "0"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", SubscriptBox["mm", "0"]]], ")"]], "2"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", SubscriptBox["mm", "0"]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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