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http://functions.wolfram.com/01.01.16.0085.01
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Sqrt[Subscript[z, 1]] Sqrt[Subscript[z, 2]] ==
-Sqrt[Subscript[z, 1] Subscript[z, 2]] /;
(Arg[Subscript[z, 1]] >= 0 && Arg[Subscript[z, 2]] >
Pi - Arg[Subscript[z, 1]]) || (Arg[Subscript[z, 1]] <= 0 &&
Arg[Subscript[z, 2]] <= -Pi - Arg[Subscript[z, 1]])
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SqrtBox[SubscriptBox["z", "1"]], " ", SqrtBox[SubscriptBox["z", "2"]]]], "\[Equal]", RowBox[List["-", SqrtBox[RowBox[List[SubscriptBox["z", "1"], " ", SubscriptBox["z", "2"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Arg", "[", SubscriptBox["z", "1"], "]"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Arg", "[", SubscriptBox["z", "2"], "]"]], ">", RowBox[List["Pi", "-", RowBox[List["Arg", "[", SubscriptBox["z", "1"], "]"]]]]]]]], "\[Or]", RowBox[List[RowBox[List[RowBox[List["Arg", "[", SubscriptBox["z", "1"], "]"]], "\[LessEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Arg", "[", SubscriptBox["z", "2"], "]"]], "\[LessEqual]", RowBox[List[RowBox[List["-", "Pi"]], "-", RowBox[List["Arg", "[", SubscriptBox["z", "1"], "]"]]]]]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msqrt> <mo> ⁢ </mo> <msqrt> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </msqrt> </mrow> <mo> ⩵ </mo> <mrow> <mo> - </mo> <msqrt> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </msqrt> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> π </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <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> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> <apply> <or /> <apply> <and /> <apply> <geq /> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <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> </apply> </apply> <apply> <and /> <apply> <leq /> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <leq /> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <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> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SqrtBox[SubscriptBox["z_", "1"]], " ", SqrtBox[SubscriptBox["z_", "2"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[SubscriptBox["zz", "1"], " ", SubscriptBox["zz", "2"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Arg", "[", SubscriptBox["zz", "1"], "]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["Arg", "[", SubscriptBox["zz", "2"], "]"]], ">", RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", SubscriptBox["zz", "1"], "]"]]]]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Arg", "[", SubscriptBox["zz", "1"], "]"]], "\[LessEqual]", "0"]], "&&", RowBox[List[RowBox[List["Arg", "[", SubscriptBox["zz", "2"], "]"]], "\[LessEqual]", RowBox[List[RowBox[List["-", "\[Pi]"]], "-", RowBox[List["Arg", "[", SubscriptBox["zz", "1"], "]"]]]]]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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