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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > Series representations > Generalized power series > Expansions at m==1





http://functions.wolfram.com/08.05.06.0065.01









  


  










Input Form





EllipticF[z, m] == (-1)^Round[Re[z]/Pi] Sin[z] Sum[((Sin[z]^(2 k) Pochhammer[1/2, k])/((1 + 2 k) k!)) Hypergeometric2F1[1/2 + k, 1 + k, 3/2 + k, Sin[z]^2] (m - 1)^k, {k, 0, Infinity}] + 2 Round[Re[z]/Pi] EllipticK[m] /; !Element[(Pi + 2 Re[z])/(4 Pi), Integers]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["2", " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], "/;", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]]










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





2007-05-02





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