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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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</ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> &#62754; </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#172; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <ms> &#960; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> &#960; </ms> </list> </apply> </apply> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <list /> <integers /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










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