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variants of this functions
EllipticPi




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Series representations > Generalized power series > Expansions at generic point m==m0 > For the function itself





http://functions.wolfram.com/08.06.06.0080.01









  


  








Input Form





EllipticPi[n, z, m] \[Proportional] EllipticPi[n, z, Subscript[m, 0]] + (1/(2 (n - Subscript[m, 0]))) (EllipticE[z, Subscript[m, 0]]/ (Subscript[m, 0] - 1) + EllipticPi[n, z, Subscript[m, 0]] - (Subscript[m, 0] Sin[2 z])/(2 (Subscript[m, 0] - 1) Sqrt[1 - Subscript[m, 0] Sin[z]^2])) (m - Subscript[m, 0]) + (1/2) (((4 Subscript[m, 0]^2 - n - Subscript[m, 0] (2 + n))/ (4 (Subscript[m, 0] - 1)^2 Subscript[m, 0] (Subscript[m, 0] - n)^2)) EllipticE[z, Subscript[m, 0]] + (1/(4 (Subscript[m, 0] - 1) Subscript[m, 0] (Subscript[m, 0] - n))) EllipticF[z, Subscript[m, 0]] + (3/(4 (Subscript[m, 0] - n)^2)) EllipticPi[n, z, Subscript[m, 0]] + ((Subscript[m, 0] - 3 Subscript[m, 0]^2 + 2 n + Subscript[m, 0] (4 Subscript[m, 0]^2 - n - Subscript[m, 0] (2 + n)) Sin[z]^2) Sin[2 z])/(8 (Subscript[m, 0] - 1)^2 (Subscript[m, 0] - n)^2 (1 - Subscript[m, 0] Sin[z]^2)^(3/2))) (m - Subscript[m, 0])^2 + \[Ellipsis] /; (m -> Subscript[m, 0])








Standard Form





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MathML Form







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</mo> <msubsup> <mi> m </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </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> &#8290; </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> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msub> <mi> m </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </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> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> EllipticPi </ci> <ci> n </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> <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> <ci> EllipticE </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <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> EllipticPi </ci> <ci> n </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> <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> <power /> <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> <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> </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> <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> <ci> EllipticE </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </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> <times /> <apply> <ci> EllipticF </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <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> <times /> <cn type='integer'> 3 </cn> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </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> <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> <times /> <apply> <plus /> <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> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </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> <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> <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'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", SubscriptBox["mm", "0"]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["EllipticE", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], RowBox[List[SubscriptBox["mm", "0"], "-", "1"]]], "+", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", SubscriptBox["mm", "0"]]], "]"]], "-", FractionBox[RowBox[List[SubscriptBox["mm", "0"], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "1"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SubscriptBox["mm", "0"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]]]], ")"]], " ", 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[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["mm", "0", "2"]]], "-", "n", "-", RowBox[List[SubscriptBox["mm", "0"], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "1"]], ")"]], "2"], " ", SubscriptBox["mm", "0"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "n"]], ")"]], "2"]]]], "+", FractionBox[RowBox[List["EllipticF", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "1"]], ")"]], " ", SubscriptBox["mm", "0"], " ", RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "n"]], ")"]]]]], "+", FractionBox[RowBox[List["3", " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", SubscriptBox["mm", "0"]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "n"]], ")"]], "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", RowBox[List["3", " ", SubsuperscriptBox["mm", "0", "2"]]], "+", RowBox[List["2", " ", "n"]], "+", RowBox[List[SubscriptBox["mm", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["mm", "0", "2"]]], "-", "n", "-", RowBox[List[SubscriptBox["mm", "0"], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "1"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["mm", "0"], "-", "n"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SubscriptBox["mm", "0"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", SubscriptBox["mm", "0"]]], ")"]]]]]]]]


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





2007-05-02