(z^m)^(1/n) == z^(m/n)/(-1)^(m/n) /; Element[m/2, Integers] || (m > 0 && Element[n, Integers]) || (n > 0 && Inequality[Pi - Pi/m, Less, Arg[z], LessEqual, Pi]) || Inequality[-Pi, Less, Arg[z], LessEqual, -Pi + Pi/m]

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "m"], ")"]], RowBox[List["1", "/", "n"]]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List["m", "/", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", FractionBox["m", "n"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["m", "/", "2"]], "\[Element]", "Integers"]], "\[Or]", RowBox[List[RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]]]], "\[Or]", RowBox[List[RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[RowBox[List["\[Pi]", "-", FractionBox["\[Pi]", "m"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]], "||", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", FractionBox["\[Pi]", "m"]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mi> m </mi> </msup> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> n </mi> </mrow> </msup> <mo> &#10869; </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> / </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mfrac> <mi> m </mi> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mi> m </mi> </mfrac> </mrow> <mo> &lt; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &lt; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mi> m </mi> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <ci> Inequality </ci> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <pi /> </apply> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <times /> <pi /> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["(", SuperscriptBox["z_", "m_"], ")"]], FractionBox["1", "n_"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["m", "/", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", FractionBox["m", "n"]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["m", "2"], "\[Element]", "Integers"]], "||", RowBox[List["(", RowBox[List[RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["n", ">", "0"]], "&&", RowBox[List[RowBox[List["\[Pi]", "-", FractionBox["\[Pi]", "m"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]], ")"]], "||", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", FractionBox["\[Pi]", "m"]]]]]]]]]]]]]

2007-05-02