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variants of this functions
EllipticE




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticE[z,m] > Series representations > Generalized power series > Expansions at m==infinity





http://functions.wolfram.com/08.04.06.0067.01









  


  








Input Form





EllipticE[z, m] == 4 Round[Re[z]/(2 Pi)] EllipticE[m] + Tan[z/2] Sqrt[1 - m Sin[z]^2] + ((2 ArcSin[Sqrt[m] Sin[z]])/(Pi Sqrt[m])) (m EllipticE[1/m] - (-1 + m) EllipticK[1/m]) - ((Sin[z] Sqrt[1 - m Sin[z]^2])/(8 m)) Sum[(Pochhammer[3/2, j]^2/(m^j ((j + 1)! (j + 2)!))) HypergeometricPFQ[{1, 1, 3/2 + j, 3/2 + j}, {3/2, 2 + j, 3 + j}, Sin[z]^2], {j, 0, Infinity}] /; (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["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["4", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["Tan", "[", FractionBox["z", "2"], "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["ArcSin", "[", RowBox[List[SqrtBox["m"], " ", RowBox[List["Sin", "[", "z", "]"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]]]], ")"]]]], RowBox[List["\[Pi]", " ", SqrtBox["m"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]], "2"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "+", "j"]], ",", RowBox[List[FractionBox["3", "2"], "+", "j"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["2", "+", "j"]], ",", RowBox[List["3", "+", "j"]]]], "}"]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["j", "+", "2"]], ")"]], "!"]]]]]]]]], RowBox[List["8", " ", "m"]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]


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





2007-05-02