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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Series representations > Generalized power series > Expansions at generic point n==n0 > For the function itself





http://functions.wolfram.com/08.03.06.0008.01









  


  










Input Form





EllipticPi[n, m] \[Proportional] EllipticPi[Subscript[n, 0], m] + (1/(2 (m - Subscript[n, 0]) (-1 + Subscript[n, 0]))) (EllipticE[m] + (-1 + m/Subscript[n, 0]) EllipticK[m] - ((m - Subscript[n, 0]^2)/Subscript[n, 0]) EllipticPi[Subscript[n, 0], m]) (n - Subscript[n, 0]) + (1/(8 (m - Subscript[n, 0])^2 (-1 + Subscript[n, 0])^2 Subscript[n, 0]^2)) ((-EllipticE[m]) Subscript[n, 0] (m + 2 m Subscript[n, 0] + (2 - 5 Subscript[n, 0]) Subscript[n, 0]) + EllipticK[m] (m^2 (1 - 4 Subscript[n, 0]) + (2 - 5 Subscript[n, 0]) Subscript[n, 0]^2 + 3 m Subscript[n, 0] (-1 + 3 Subscript[n, 0])) + EllipticPi[Subscript[n, 0], m] (2 m (2 - 5 Subscript[n, 0]) Subscript[n, 0] + 3 Subscript[n, 0]^4 + m^2 (-1 + 4 Subscript[n, 0]))) (n - Subscript[n, 0])^2 + \[Ellipsis] /; (n -> Subscript[n, 0])










Standard Form





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MathML Form







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</apply> </apply> </apply> <apply> <times /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> 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<power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </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> n </ci> <apply> <ci> Subscript </ci> <ci> n </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_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List[SubscriptBox["nn", "0"], ",", "m"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["m", SubscriptBox["nn", "0"]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", "-", SubsuperscriptBox["nn", "0", "2"]]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List[SubscriptBox["nn", "0"], ",", "m"]], "]"]]]], SubscriptBox["nn", "0"]]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", SubscriptBox["nn", "0"]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", SubscriptBox["nn", "0"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["nn", "0"]]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", "m", "]"]]]], " ", SubscriptBox["nn", "0"], " ", RowBox[List["(", RowBox[List["m", "+", RowBox[List["2", " ", "m", " ", SubscriptBox["nn", "0"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", SubscriptBox["nn", "0"]]]]], ")"]], " ", SubscriptBox["nn", "0"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SubscriptBox["nn", "0"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", SubscriptBox["nn", "0"]]]]], ")"]], " ", SubsuperscriptBox["nn", "0", "2"]]], "+", RowBox[List["3", " ", "m", " ", SubscriptBox["nn", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["3", " ", SubscriptBox["nn", "0"]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List[SubscriptBox["nn", "0"], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", SubscriptBox["nn", "0"]]]]], ")"]], " ", SubscriptBox["nn", "0"]]], "+", RowBox[List["3", " ", SubsuperscriptBox["nn", "0", "4"]]], "+", RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SubscriptBox["nn", "0"]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", SubscriptBox["nn", "0"]]], ")"]], "2"]]], RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["nn", "0"]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["nn", "0"]]], ")"]], "2"], " ", SubsuperscriptBox["nn", "0", "2"]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", SubscriptBox["nn", "0"]]], ")"]]]]]]]]










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





2007-05-02





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