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variants of this functions
EllipticE






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticE[z] > Series representations > Generalized power series > Expansions on branch cuts > For the function itself





http://functions.wolfram.com/08.01.06.0020.01









  


  










Input Form





EllipticE[z] \[Proportional] -2 I (-1)^Floor[Arg[-z + x]/(2 Pi)] (EllipticE[1 - x] - EllipticK[1 - x]) Floor[Arg[-z + x]/(2 Pi)] + EllipticE[x] + (1/(2 x)) (EllipticE[x] - EllipticK[x] - 2 I (-1)^Floor[Arg[-z + x]/(2 Pi)] Floor[Arg[-z + x]/(2 Pi)] EllipticE[1 - x]) (z - x) + (1/(4 Pi)) (((Pi I)/((-1 + x) x^2)) (-1)^Floor[Arg[-z + x]/(2 Pi)] Floor[Arg[-z + x]/(2 Pi)] (-2 EllipticE[1 - x] + (EllipticE[1 - x] + EllipticK[1 - x]) x) - MeijerG[{{-(1/2), -(3/2)}, {}}, {{0, -1}, {}}, 1 - x]) (z - x)^2 + \[Ellipsis] /; (z -> x) && Element[x, Reals] && x > 1










Standard Form





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MathML Form







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<ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticE </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <exponentiale /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <imaginaryi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> 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</apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus 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<cn type='integer'> 0 </cn> <cn type='integer'> -1 </cn> </list> <list /> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <ci> x </ci> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]], "-", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["EllipticE", "[", "x", "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "x", "]"]], "-", RowBox[List["EllipticK", "[", "x", "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], RowBox[List["2", " ", "x"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]], "+", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]]]], ")"]], " ", "x"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], " ", SuperscriptBox["x", "2"]]]], "-", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "1"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", "x"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], RowBox[List["4", " ", "\[Pi]"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "1"]]]]]]]]]]










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





2007-05-02





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