DirectedInfinity[E^(I x)] == E^(I x) Infinity /; Element[x, Reals]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DirectedInfinity", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "x"]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "x"]]], "\[Infinity]"]]]], "/;", " ", RowBox[List["x", "\[Element]", "Reals"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> </apply> <infinity /> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> </apply> <infinity /> </apply> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DirectedInfinity", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "x_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "x"]]], " ", "\[Infinity]"]], "/;", RowBox[List["x", "\[Element]", "Reals"]]]]]]]]

2001-10-29