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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0102.01

 Input Form

 ChebyshevT[\[Nu], z] == ((2^(-2 - 3 \[Nu]) (-1 + z)^\[Nu])/ (Pi Gamma[-2 \[Nu]])) (Sum[Residue[((Gamma[s] Gamma[-s - \[Nu]] Gamma[1/2 - s - \[Nu]])/ ((z + 1)/(z - 1))^s) Gamma[1/2 + s], {s, -(1/2) - j}], {j, 0, Infinity}] + Sum[Residue[((Gamma[1/2 + s] Gamma[-s - \[Nu]] Gamma[1/2 - s - \[Nu]])/ ((z + 1)/(z - 1))^s) Gamma[s], {s, -j}], {j, 0, Infinity}]) + (2^(-2 + 3 \[Nu])/((-1 + z)^\[Nu] (Pi Gamma[2 \[Nu]]))) (Sum[Residue[((Gamma[s] Gamma[-s + \[Nu]] Gamma[1/2 - s + \[Nu]])/ ((z + 1)/(z - 1))^s) Gamma[1/2 + s], {s, -(1/2) - j}], {j, 0, Infinity}] + Sum[Residue[((Gamma[1/2 + s] Gamma[-s + \[Nu]] Gamma[1/2 - s + \[Nu]])/ ((z + 1)/(z - 1))^s) Gamma[s], {s, -j}], {j, 0, Infinity}]) /; Abs[(z + 1)/(z - 1)] < 1 && !Element[2 \[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["3", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "\[Nu]"]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", " ", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]], "+", RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], 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[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Nu]"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]]]], " ", ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", RowBox[List["j", ",", "0", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]], "+", RowBox[List["Sum", "[", RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], 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[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], "]"]], "<", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", "\[Nu]"]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 T ν ( z ) 2 - 3 ν - 2 ( z - 1 ) ν π Γ ( - 2 ν ) ( j = 0 res s ( ( Γ ( s + 1 2 ) Γ ( - s - ν ) Γ ( - s - ν + 1 2 ) ( z + 1 z - 1 ) - s ) Γ ( s ) ) ( - j ) + j = 0 res s ( ( Γ ( s ) Γ ( - s - ν ) Γ ( - s - ν + 1 2 ) ( z + 1 z - 1 ) - s ) Γ ( s + 1 2 ) ) ( - j - 1 2 ) ) + 2 3 ν - 2 ( z - 1 ) - ν π Γ ( 2 ν ) ( j = 0 res s ( ( Γ ( s + 1 2 ) Γ ( ν - s ) Γ ( - s + ν + 1 2 ) ( z + 1 z - 1 ) - s ) Γ ( s ) ) ( - j ) + j = 0 res s ( ( Γ ( s ) Γ ( ν - s ) Γ ( - s + ν + 1 2 ) ( z + 1 z - 1 ) - s ) Γ ( s + 1 2 ) ) ( - j - 1 2 ) ) /; "\[LeftBracketingBar]" z + 1 z - 1 "\[RightBracketingBar]" < 1 2 ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition ChebyshevT ν z 2 -3 ν -2 z -1 ν Gamma -2 ν -1 j 0 DirectedInfinity 1 Subscript res s Gamma s 1 2 Gamma -1 s -1 ν Gamma -1 s -1 ν 1 2 z 1 z -1 -1 -1 s Gamma s -1 j j 0 DirectedInfinity 1 Subscript res s Gamma s Gamma -1 s -1 ν Gamma -1 s -1 ν 1 2 z 1 z -1 -1 -1 s Gamma s 1 2 -1 j -1 1 2 2 3 ν -2 z -1 -1 ν Gamma 2 ν -1 j 0 DirectedInfinity 1 Subscript res s Gamma s 1 2 Gamma ν -1 s Gamma -1 s ν 1 2 z 1 z -1 -1 -1 s Gamma s -1 j j 0 DirectedInfinity 1 Subscript res s Gamma s Gamma ν -1 s Gamma -1 s ν 1 2 z 1 z -1 -1 -1 s Gamma s 1 2 -1 j -1 1 2 z 1 z -1 -1 1 2 ν [/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[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["3", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "\[Nu]"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "\[Element]", "Integers"]]]]]]]]]]]]

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

 2007-05-02