Sqrt[DirectedInfinity[z]] == DirectedInfinity[Sqrt[Sign[z]]]

Cell[BoxData[RowBox[List[SqrtBox[RowBox[List["DirectedInfinity", "[", "z", "]"]]], "\[Equal]", RowBox[List["DirectedInfinity", "[", SqrtBox[RowBox[List["Sign", "[", "z", "]"]]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msqrt> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> &#10869; </mo> <mrow> <msqrt> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </msqrt> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <times /> <ci> z </ci> <infinity /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Sign </ci> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SqrtBox[RowBox[List["DirectedInfinity", "[", "z_", "]"]]], "]"]], "\[RuleDelayed]", RowBox[List["DirectedInfinity", "[", SqrtBox[RowBox[List["Sign", "[", "z", "]"]]], "]"]]]]]]

2001-10-29