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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 n==infinity





http://functions.wolfram.com/08.03.06.0028.01









  


  










Input Form





EllipticPi[n, m] \[Proportional] (Pi/(2 Sqrt[-n])) (1 + (1 + m)/(2 n) + (3 + 2 m + 3 m^2)/(8 n^2) + O[1/n^3]) + (1/(4 n)) (4 (EllipticE[m] - EllipticK[m]) - (-8 (1 + m) EllipticE[m] + 4 (2 + m) EllipticK[m])/(3 n) + (4 ((8 + 7 m + 8 m^2) EllipticE[m] - (8 + 3 m + 4 m^2) EllipticK[m]))/ (15 n^2) + O[1/n^3])










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "+", "m"]], RowBox[List["2", " ", "n"]]], "+", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["8", " ", SuperscriptBox["n", "2"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["n", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "3"]], "]"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "n"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], "-", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["2", "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], RowBox[List["3", " ", "n"]]], "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["8", "+", RowBox[List["7", " ", "m"]], "+", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["8", "+", RowBox[List["3", " ", "m"]], "+", RowBox[List["4", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], RowBox[List["15", " ", SuperscriptBox["n", "2"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["n", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "3"]], "]"]]]], RowBox[List["4", " ", "n"]]]]]]]]]










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





2007-05-02





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