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http://functions.wolfram.com/01.02.16.0104.01
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(c E^z)^a == c^a E^(a z) Exp[2 Pi I a Floor[(Pi - Arg[c] - Im[z])/(2 Pi)]]
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Cell[BoxData[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", "z"]]], ")"]], "a"], "\[Equal]", RowBox[List[SuperscriptBox["c", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], " ", RowBox[List["Exp", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", "c", "]"]], "-", RowBox[List["Im", "[", "z", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]]]]]]
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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> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mi> z </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⩵ </mo> <mrow> <msup> <mi> c </mi> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <apply> <power /> <ci> c </ci> <ci> a </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> a </ci> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <ci> c </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <ci> z </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> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["(", RowBox[List["c_", " ", SuperscriptBox["\[ExponentialE]", "z_"]]], ")"]], "a_"], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["c", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", "c", "]"]], "-", RowBox[List["Im", "[", "z", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
