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http://functions.wolfram.com/01.33.27.0002.01
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Root[Function[z, z^n - a], n] == a^(1/n) /;
n == 2 || ((Element[n - 3, Integers] && n - 3 >= 0) &&
!IntervalMemberQ[Interval[{0, Infinity}], a])
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Root", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List["z", ",", RowBox[List[SuperscriptBox["z", "n"], "-", "a"]]]], "]"]], ",", "n"]], "]"]], "\[Equal]", SuperscriptBox["a", RowBox[List["1", "/", "n"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", "2"]], "||", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["n", "-", "3"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "3"]], "\[GreaterEqual]", "0"]]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "\[Infinity]"]], "}"]], "]"]], ",", "a"]], "]"]], "]"]]]], ")"]]]]]]]]
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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> <msubsup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⩵ </mo> <msup> <mi> a </mi> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ⩵ </mo> <mn> 2 </mn> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mtext> ​ </mtext> <mrow> <mi> a </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msubsup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⩵ </mo> <msup> <mi> a </mi> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ⩵ </mo> <mn> 2 </mn> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mtext> ​ </mtext> <mrow> <mi> a </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Root", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List["z_", ",", RowBox[List[SuperscriptBox["z_", "n_"], "-", "a_"]]]], "]"]], ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["a", RowBox[List["1", "/", "n"]]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", "2"]], "||", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["n", "-", "3"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "3"]], "\[GreaterEqual]", "0"]]]], ")"]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "\[Infinity]"]], "}"]], "]"]], ",", "a"]], "]"]]]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
