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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/08.03.06.0029.01









  


  










Input Form





EllipticPi[n, m] == (-((Pi m)/(4 n))) Sum[((Pochhammer[3/2, k]/(k + 1)!) m^k Hypergeometric2F1[1/2, 3/2 + k, 2 + k, m])/n^k, {k, 0, Infinity}] + (Pi/(2 Sqrt[-n])) Sum[((Pochhammer[1/2, k]/k!) Hypergeometric2F1[1/2, -k, 1/2 - k, m])/n^k, {k, 0, Infinity}]










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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", "m"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["m", "k"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", RowBox[List["2", "+", "k"]], ",", "m"]], "]"]], " ", SuperscriptBox["n", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]]]], RowBox[List["4", " ", "n"]]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", "m"]], "]"]], " ", SuperscriptBox["n", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "n"]]]]]]]]]]]]










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





2007-05-02





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