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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0083.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Nu]", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], 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[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], 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[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], 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[" ", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], SqrtBox[RowBox[List["1", "+", "z"]]], 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[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], 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[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], 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["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], "<", "2"]]]]]]

 MathML Form

 T ν ( z ) cos ( π ν ) sin 2 ( π ν ) 2 π 3 / 2 z + 1 ( j = 0 res s ( ( Γ ( s + 1 2 ) Γ ( - s - ν + 1 2 ) Γ ( - s + ν + 1 2 ) ( 1 - z 2 ) - s ) Γ ( s ) ) ( - j ) + j = 0 res s ( ( Γ ( s ) Γ ( - s - ν + 1 2 ) Γ ( - s + ν + 1 2 ) ( 1 - z 2 ) - s ) Γ ( s + 1 2 ) ) ( - j - 1 2 ) ) - ν cos 2 ( π ν ) sin ( π ν ) π 3 / 2 ( j = 0 res s ( ( Γ ( s + 1 2 ) Γ ( - s - ν ) Γ ( ν - s ) ( 1 - z 2 ) - s ) Γ ( s ) ) ( - j ) + j = 0 res s ( ( Γ ( s ) Γ ( - s - ν ) Γ ( ν - s ) ( 1 - z 2 ) - s ) Γ ( s + 1 2 ) ) ( - j - 1 2 ) ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 2 Condition ChebyshevT ν z ν ν 2 2 1 2 3 2 -1 z 1 1 2 j 0 DirectedInfinity 1 Subscript res s Gamma s 1 2 Gamma -1 s -1 ν 1 2 Gamma -1 s ν 1 2 1 -1 z 2 -1 -1 s Gamma s -1 j j 0 DirectedInfinity 1 Subscript res s Gamma s Gamma -1 s -1 ν 1 2 Gamma -1 s ν 1 2 1 -1 z 2 -1 -1 s Gamma s 1 2 -1 j -1 1 2 -1 ν ν 2 ν 3 2 -1 j 0 DirectedInfinity 1 Subscript res s Gamma s 1 2 Gamma -1 s -1 ν Gamma ν -1 s 1 -1 z 2 -1 -1 s Gamma s -1 j j 0 DirectedInfinity 1 Subscript res s Gamma s Gamma -1 s -1 ν Gamma ν -1 s 1 -1 z 2 -1 -1 s Gamma s 1 2 -1 j -1 1 2 1 -1 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[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[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], 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[RowBox[List["-", "s"]], "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]], SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]], ")"]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", 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[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], 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[FractionBox["1", "2"], "-", "s", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], "<", "2"]]]]]]]]

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

 2007-05-02