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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0099.01

 Input Form

 ChebyshevT[\[Nu], z] == ((Sqrt[1 + z] \[Nu] Sin[Pi \[Nu]])/(2 Sqrt[2 Pi])) (2^(1/2 - \[Nu]) (-1 - z)^(-(1/2) + \[Nu]) Sum[Residue[((Gamma[-s - \[Nu]] Gamma[1/2 - s - \[Nu]])/ ((-(2/(1 + z)))^s Gamma[1 - s - 2 \[Nu]])) Gamma[s], {s, -j}], {j, 0, Infinity}] - 2^(1/2 + \[Nu]) (-1 - z)^(-(1/2) - \[Nu]) Sum[Residue[((Gamma[-s + \[Nu]] Gamma[1/2 - s + \[Nu]])/ ((-(2/(1 + z)))^s Gamma[1 - s + 2 \[Nu]])) Gamma[s], {s, -j}], {j, 0, Infinity}]) + ((\[Nu] Cos[Pi \[Nu]])/(2 Sqrt[Pi])) ((-2^(-\[Nu])) (-1 - z)^\[Nu] Sum[Residue[((Gamma[-s - \[Nu]] Gamma[1/2 - s - \[Nu]])/ ((-(2/(1 + z)))^s Gamma[1 - s - 2 \[Nu]])) Gamma[s], {s, -j}], {j, 0, Infinity}] - (2^\[Nu] Sum[Residue[((Gamma[-s + \[Nu]] Gamma[1/2 - s + \[Nu]])/ ((-(2/(1 + z)))^s Gamma[1 - s + 2 \[Nu]])) Gamma[s], {s, -j}], {j, 0, Infinity}])/(-1 - z)^\[Nu]) /; Abs[z + 1] > 2

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", "z"]]], " ", "\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], " ", RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], "\[Nu]"], RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "+", "1"]], "]"]], ">", "2"]]]]]]

 MathML Form

 T ν ( z ) z + 1 ν sin ( π ν ) 2 2 π ( 2 1 2 - ν ( - z - 1 ) ν - 1 2 j = 0 res s ( Γ ( - s - ν ) Γ ( - s - ν + 1 2 ) ( - 2 z + 1 ) - s Γ ( - s - 2 ν + 1 ) Γ ( s ) ) ( - j ) - 2 ν + 1 2 ( - z - 1 ) - ν - 1 2 j = 0 res s ( Γ ( ν - s ) Γ ( - s + ν + 1 2 ) ( - 2 z + 1 ) - s Γ ( - s + 2 ν + 1 ) Γ ( s ) ) ( - j ) ) + ν cos ( π ν ) 2 π ( - 2 ν ( - z - 1 ) - ν j = 0 res s ( Γ ( ν - s ) Γ ( - s + ν + 1 2 ) ( - 2 z + 1 ) - s Γ ( - s + 2 ν + 1 ) Γ ( s ) ) ( - j ) - 2 - ν ( - z - 1 ) ν j = 0 res s ( Γ ( - s - ν ) Γ ( - s - ν + 1 2 ) ( - 2 z + 1 ) - s Γ ( - s - 2 ν + 1 ) Γ ( s ) ) ( - j ) ) /; "\[LeftBracketingBar]" z + 1 "\[RightBracketingBar]" > 2 Condition ChebyshevT ν z z 1 1 2 ν ν 2 2 1 2 -1 2 1 2 -1 ν -1 z -1 ν -1 1 2 j 0 DirectedInfinity 1 Subscript res s Gamma -1 s -1 ν Gamma -1 s -1 ν 1 2 -1 2 z 1 -1 -1 s Gamma -1 s -1 2 ν 1 -1 Gamma s -1 j -1 2 ν 1 2 -1 z -1 -1 ν -1 1 2 j 0 DirectedInfinity 1 Subscript res s Gamma ν -1 s Gamma -1 s ν 1 2 -1 2 z 1 -1 -1 s Gamma -1 s 2 ν 1 -1 Gamma s -1 j ν ν 2 1 2 -1 -1 2 ν -1 z -1 -1 ν j 0 DirectedInfinity 1 Subscript res s Gamma ν -1 s Gamma -1 s ν 1 2 -1 2 z 1 -1 -1 s Gamma -1 s 2 ν 1 -1 Gamma s -1 j -1 2 -1 ν -1 z -1 ν j 0 DirectedInfinity 1 Subscript res s Gamma -1 s -1 ν Gamma -1 s -1 ν 1 2 -1 2 z 1 -1 -1 s Gamma -1 s -1 2 ν 1 -1 Gamma s -1 j z 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", "z"]]], " ", "\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "-", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]], "-", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["2", RowBox[List["1", "+", "z"]]]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "+", "1"]], "]"]], ">", "2"]]]]]]]]

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

 2007-05-02