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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Differentiation > Low-order differentiation > With respect to n





http://functions.wolfram.com/08.03.20.0002.01









  


  










Input Form





D[EllipticPi[n, m], {n, 2}] == ((m (1 - 4 n) + n (5 n - 2))/(4 (m - n) (n - 1)^2 n^2)) EllipticK[m] - ((m (1 + 2 n) + (2 - 5 n) n)/(4 (m - n)^2 (n - 1)^2 n)) EllipticE[m] + ((2 m (2 - 5 n) n + 3 n^4 + m^2 (4 n - 1))/(4 (m - n)^2 (n - 1)^2 n^2)) EllipticPi[n, m]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["n", ",", "2"]], "}"]]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["m", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "n"]]]], ")"]]]], "+", RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "n"]], "-", "2"]], ")"]]]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"], " ", SuperscriptBox["n", "2"]]]], RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["m", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", "n"]]]], ")"]], " ", "n"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"], " ", "n"]]], RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", "n"]]]], ")"]], " ", "n"]], "+", RowBox[List["3", " ", SuperscriptBox["n", "4"]]], "+", RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "n"]], "-", "1"]], ")"]]]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"], " ", SuperscriptBox["n", "2"]]]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["n_", ",", "2"]], "}"]]]]], RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "n"]]]], ")"]]]], "+", RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", "n"]], "-", "2"]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"], " ", SuperscriptBox["n", "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", "n"]]]], ")"]], " ", "n"]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"], " ", "n"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", "n"]]]], ")"]], " ", "n"]], "+", RowBox[List["3", " ", SuperscriptBox["n", "4"]]], "+", RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "n"]], "-", "1"]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"], " ", SuperscriptBox["n", "2"]]]]]]]]]]










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





2001-10-29





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