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http://functions.wolfram.com/07.05.06.0083.01
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ChebyshevU[\[Nu], z] \[Proportional]
Piecewise[{{(1/2 - (I z)/(2 Sqrt[1 - z^2])) E^(I \[Nu] ArcCos[z]),
-Pi < Arg[(\[Nu] + 1) ArcCos[z]] < 0},
{(1/2 + (I z)/(2 Sqrt[1 - z^2]))/E^(I \[Nu] ArcCos[z]),
0 < Arg[(\[Nu] + 1) ArcCos[z]] < Pi}}, Sin[(\[Nu] + 1) ArcCos[z]]/
Sqrt[1 - z^2]] /; (Abs[\[Nu]] -> Infinity)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]], ",", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], "<", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Nu]", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]], ",", RowBox[List["0", "<", RowBox[List["Arg", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], "<", "\[Pi]"]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]
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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> <mrow> <msub> <mi> U </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ν </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> ChebyshevU </ci> <ci> ν </ci> <ci> z </ci> </apply> <piecewise> <piece> <apply> <times /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> ν </ci> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <arg /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> ν </ci> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <arg /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> <pi /> </apply> </piece> <otherwise> <apply> <times /> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> ν </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]], RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], "<", "0"]]], List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Nu]", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]]], RowBox[List["0", "<", RowBox[List["Arg", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], "<", "\[Pi]"]]], List[FractionBox[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
