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http://functions.wolfram.com/01.02.06.0044.01
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z^a == E^(2 I a Pi Floor[Arg[z - x]/(2 Pi)]) x^a (z/x)^a /;
Abs[z/x - 1] < 1 && Element[x, Reals] && x < 0
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["z", "a"], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox["x", "a"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "x"], ")"]], "a"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List[FractionBox["z", "x"], "-", "1"]], "]"]], "<", "1"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]
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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> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo>  </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> x </mi> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mi> x </mi> </mfrac> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mfrac> <mi> z </mi> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> a </ci> <pi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </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> <power /> <ci> x </ci> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> a </ci> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["z_", "a_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox["x", "a"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "x"], ")"]], "a"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List[FractionBox["z", "x"], "-", "1"]], "]"]], "<", "1"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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