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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 m==infinity





http://functions.wolfram.com/08.06.06.0103.01









  


  










Input Form





EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Sqrt[(-m) Sin[z]^2]/(2 m Sin[z])) (-Log[-4 m Sin[z]^2] + (2 Log[Cos[z/2]^2] - ((2 Sqrt[n])/Sqrt[-1 + n]) (ArcTanh[(Cos[z] Sqrt[n])/Sqrt[-1 + n]] - ArcTanh[Sqrt[n]/Sqrt[-1 + n]])) (1 + O[1/m])) + 2 Round[Re[z]/Pi] ((Log[-m]/(2 Sqrt[-m])) (1 + O[1/m]) + ((Sqrt[n] ArcSin[Sqrt[n]])/(Sqrt[1 - n] Sqrt[-m])) (1 + O[1/m]) + ((4 Log[2])/(2 Sqrt[-m])) (1 + O[1/m])) /; (Abs[m] -> 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_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["Cos", "[", FractionBox["z", "2"], "]"]], "2"], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["n"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", SqrtBox["n"]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], "-", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox["n"], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "m"], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "m", " ", RowBox[List["Sin", "[", "z", "]"]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "m"], "]"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["n"], " ", RowBox[List["ArcSin", "[", SqrtBox["n"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "m"], "]"]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", SqrtBox[RowBox[List["-", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "m"], "]"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "m"]]]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02