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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0044.01

 Input Form

 ChebyshevT[\[Nu], z] \[Proportional] Sin[Pi \[Nu]] Sqrt[1 - x^2] Exp[Pi I Floor[Arg[z - x]/(2 Pi)]] ChebyshevU[\[Nu] - 1, -x] + Cos[Pi \[Nu]] (-2 I I^Floor[Arg[z - x]/(2 Pi)] Floor[Arg[z - x]/(2 Pi)] + Exp[Pi I Floor[Arg[z - x]/(2 Pi)]]) ChebyshevT[\[Nu], -x] + (((\[Nu] Sin[Pi \[Nu]])/Sqrt[1 - x^2]) Exp[Pi I Floor[Arg[z - x]/(2 Pi)]] ChebyshevT[\[Nu], -x] - \[Nu] Cos[Pi \[Nu]] (-2 I I^Floor[Arg[z - x]/(2 Pi)] Floor[Arg[z - x]/(2 Pi)] + Exp[Pi I Floor[Arg[z - x]/(2 Pi)]]) ChebyshevU[\[Nu] - 1, -x]) (z - x) + O[(z - x)^2] /; Element[x, Reals] && x < -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", RowBox[List["-", "x"]]]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], ")"]], RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "x"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], " ", RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "x"]]]], "]"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", RowBox[List["-", "x"]]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 T ν ( z ) cos ( π ν ) ( - 2 arg ( z - x ) 2 π arg ( z - x ) 2 π + π arg ( z - x ) 2 π ) T ν ( - x ) + 1 - x 2 π arg ( z - x ) 2 π sin ( π ν ) U ν - 1 ( - x ) + ( ν sin ( π ν ) 1 - x 2 π arg ( z - x ) 2 π T ν ( - x ) - ν cos ( π ν ) ( - 2 arg ( z - x ) 2 π arg ( z - x ) 2 π + π arg ( z - x ) 2 π ) U ν - 1 ( - x ) ) ( z - x ) + O ( ( z - x ) 2 ) /; x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] x < - 1 Condition Proportional ChebyshevT ν z ν -2 z -1 x 2 -1 z -1 x 2 -1 z -1 x 2 -1 ChebyshevT ν -1 x 1 -1 x 2 1 2 z -1 x 2 -1 ν ChebyshevU ν -1 -1 x ν ν 1 -1 x 2 1 2 -1 z -1 x 2 -1 ChebyshevT ν -1 x -1 ν ν -2 z -1 x 2 -1 z -1 x 2 -1 z -1 x 2 -1 ChebyshevU ν -1 -1 x z -1 x O z -1 x 2 x x -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", RowBox[List["-", "x"]]]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "x"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "x"]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", RowBox[List["-", "x"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", "x"]], "]"]], "2"]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", RowBox[List["-", "1"]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02