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variants of this functions
EllipticE






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticE[z,m] > Series representations > Generalized power series > Expansions on branch cuts > Formulas for vertical intervals > For Re(z0/2 Pi-1/4) ∈ Z





http://functions.wolfram.com/08.04.06.0021.01









  


  










Input Form





EllipticE[z, m] \[Proportional] (2 (2 Re[Subscript[z, 0]/(2 Pi) - 1/4] + 1) EllipticE[m] - Sqrt[m] (EllipticE[1/m] + (-1 + 1/m) EllipticK[1/m])) (1 + Exp[(-Pi) I (Floor[3/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[3/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])]) - Exp[(-Pi) I (Floor[3/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[3/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])] EllipticE[Subscript[z, 0], m] + Sqrt[1 - m Sin[Subscript[z, 0]]^2] (z - Subscript[z, 0]) - ((m Sin[2 Subscript[z, 0]])/ (4 Sqrt[1 - m Sin[Subscript[z, 0]]^2])) (z - Subscript[z, 0])^2 + \[Ellipsis] /; (z -> Subscript[z, 0]) && Element[Re[Subscript[z, 0]/(2 Pi) - 1/4], Integers]










Standard Form





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MathML Form







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<power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <in /> <apply> <real /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["zz", "0"], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox["1", "4"]]], "]"]]]], "+", "1"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[SqrtBox["m"], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "m"]]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[SubscriptBox["zz", "0"], ",", "m"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SubscriptBox["zz", "0"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]]]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["zz", "0"], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox["1", "4"]]], "]"]], "\[Element]", "Integers"]]]]]]]]]]










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





2007-05-02





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