Arg[ProductLog[x + I y]] == Sum[((-k)^(k - 1)/k!) ArcTan[Cos[k ArcTan[x, y]], Sin[k ArcTan[x, y]]], {k, 1, Infinity}] /; x^2 + y^2 < 1/E^2

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Arg", "[", RowBox[List["ProductLog", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "k"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List["k", "!"]]], RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Cos", "[", RowBox[List["k", " ", RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]]]], "]"]], ",", RowBox[List["Sin", "[", RowBox[List["k", " ", RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]]]], "]"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "<", FractionBox["1", SuperscriptBox["\[ExponentialE]", "2"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> W </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> &lt; </mo> <mfrac> <mn> 1 </mn> <msup> <mi> &#8519; </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arg /> <apply> <ci> W </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <apply> <cos /> <apply> <times /> <ci> k </ci> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> k </ci> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <exponentiale /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Arg", "[", RowBox[List["ProductLog", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "k"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Cos", "[", RowBox[List["k", " ", RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]]]], "]"]], ",", RowBox[List["Sin", "[", RowBox[List["k", " ", RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]]]], "]"]]]], "]"]]]], RowBox[List["k", "!"]]]]], "/;", RowBox[List[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "<", FractionBox["1", SuperscriptBox["\[ExponentialE]", "2"]]]]]]]]]]

2001-10-29