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http://functions.wolfram.com/01.03.16.0035.01
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E^(I n ArcTan[x, y]) == ChebyshevT[n, x/Sqrt[x^2 + y^2]] +
((I y)/Sqrt[x^2 + y^2]) ChebyshevU[n - 1, x/Sqrt[x^2 + y^2]] /;
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "n", " ", RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]]]]], "\[Equal]", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", FractionBox["x", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]]], "]"]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "y"]], SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", FractionBox["x", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "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> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⩵ </mo> <mrow> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mi> x </mi> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mfrac> <mi> x </mi> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> n </ci> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <ci> y </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ChebyshevU </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "n_", " ", RowBox[List["ArcTan", "[", RowBox[List["x_", ",", "y_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", FractionBox["x", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "y"]], ")"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", FractionBox["x", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]]], "]"]]]], SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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