Sqrt[Subscript[z, 1] Subscript[z, 2]] == Sqrt[Subscript[z, 1]] Sqrt[Subscript[z, 2]] Exp[Pi I Floor[(Pi - Arg[Subscript[z, 1]] - Arg[Subscript[z, 2]])/(2 Pi)]]

Cell[BoxData[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["z", "1"], " ", SubscriptBox["z", "2"]]]], "\[Equal]", RowBox[List[SqrtBox[SubscriptBox["z", "1"]], " ", SqrtBox[SubscriptBox["z", "2"]], RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", SubscriptBox["z", "1"], "]"]], "-", RowBox[List["Arg", "[", SubscriptBox["z", "2"], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msqrt> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </msqrt> <mo> &#10869; </mo> <mrow> <msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msqrt> <mo> &#8290; </mo> <msqrt> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </msqrt> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <exp /> <apply> <times /> <pi /> <imaginaryi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SqrtBox[RowBox[List[SubscriptBox["z_", "1"], " ", SubscriptBox["z_", "2"]]]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[SubscriptBox["zz", "1"]], " ", SqrtBox[SubscriptBox["zz", "2"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", SubscriptBox["zz", "1"], "]"]], "-", RowBox[List["Arg", "[", SubscriptBox["zz", "2"], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]]

2001-10-29