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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/08.05.06.0063.01









  


  










Input Form





EllipticF[z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Log[Sec[z] + Tan[z]] + (1/4) (Sec[z] Tan[z] - Log[Sec[z] + Tan[z]]) (m - 1) + (3/256) (12 Log[Sec[z] + Tan[z]] - 5 Sec[z]^4 Sin[3 z] + 3 Sec[z]^3 Tan[z]) (m - 1)^2 + \[Ellipsis]) - Round[Re[z]/Pi] ((1 - (m - 1)/4 + (9/64) (m - 1)^2 + \[Ellipsis]) Log[1 - m] - 4 Log[2] + (1/2) (m - 1) (Log[4] - 1) - (9/32) (m - 1)^2 (-(7/6) + Log[4]) + \[Ellipsis]) /; (m -> 1) && !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["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["Sec", "[", "z", "]"]], "+", RowBox[List["Tan", "[", "z", "]"]]]], "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sec", "[", "z", "]"]], " ", RowBox[List["Tan", "[", "z", "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["Sec", "[", "z", "]"]], "+", RowBox[List["Tan", "[", "z", "]"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["3", "256"], " ", RowBox[List["(", RowBox[List[RowBox[List["12", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["Sec", "[", "z", "]"]], "+", RowBox[List["Tan", "[", "z", "]"]]]], "]"]]]], "-", RowBox[List["5", " ", SuperscriptBox[RowBox[List["Sec", "[", "z", "]"]], "4"], " ", RowBox[List["Sin", "[", RowBox[List["3", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Sec", "[", "z", "]"]], "3"], " ", RowBox[List["Tan", "[", "z", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "-", RowBox[List[RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["m", "-", "1"]], "4"], "+", RowBox[List[FractionBox["9", "64"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]]]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox["9", "32"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "+", RowBox[List["Log", "[", "4", "]"]]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]], "&&", 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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