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http://functions.wolfram.com/12.05.16.0018.01
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Conjugate[z^a] == Abs[z]^a (Cos[a ArcTan[Re[z], Im[z]]] -
I Sin[a ArcTan[Re[z], Im[z]]]) /; Element[a, Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Conjugate", "[", SuperscriptBox["z", "a"], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], "a"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List["a", ",", "Reals"]], "]"]]]]]]
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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> <mover> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> _ </mo> </mover> <mo> ⩵ </mo> <mrow> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> OverBar </ci> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <abs /> <ci> z </ci> </apply> <ci> a </ci> </apply> <apply> <plus /> <apply> <cos /> <apply> <times /> <ci> a </ci> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <times /> <ci> a </ci> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Conjugate", "[", SuperscriptBox["z_", "a_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], "a"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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