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 ChebyshevT

 http://functions.wolfram.com/07.04.06.0082.01

 Input Form

 ChebyshevT[\[Nu], z] == (-((\[Nu] Cos[(Pi \[Nu])/2] Sin[Pi \[Nu]])/(4 Pi^(3/2)))) (Sum[Residue[((Gamma[s] Gamma[-s - \[Nu]/2] Gamma[-s + \[Nu]/2])/ (1 - z^2)^s) Gamma[1/2 + s], {s, -(1/2) - j}], {j, 0, Infinity}] + Sum[Residue[((Gamma[1/2 + s] Gamma[-s - \[Nu]/2] Gamma[-s + \[Nu]/2])/ (1 - z^2)^s) Gamma[s], {s, -j}], {j, 0, Infinity}]) + ((z Sin[(Pi \[Nu])/2] Sin[Pi \[Nu]])/(2 Pi^(3/2))) (Sum[Residue[(Gamma[-s + (1 - \[Nu])/2] Gamma[1/2 + s] Gamma[-s + (1 + \[Nu])/2])/(1 - z^2)^s, {s, -j}], {j, 0, Infinity}] + Sum[Residue[((Gamma[s] Gamma[-s + (1 - \[Nu])/2] Gamma[-s + (1 + \[Nu])/2])/(1 - z^2)^s) Gamma[1/2 + s], {s, -(1/2) - j}], {j, 0, Infinity}]) /; Abs[1 - z^2] < 1

 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]", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["4", " ", 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"]], "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox["\[Nu]", "2"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["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"]], "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox["\[Nu]", "2"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["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["z", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", 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["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]]]], " ", ")"]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "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", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["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[RowBox[List["Abs", "[", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], "]"]], "<", "1"]]]]]]

 MathML Form

 T ν ( z ) z sin ( π ν 2 ) sin ( π ν ) 2 π 3 / 2 ( j = 0 res s ( Γ ( 1 - ν 2 - s ) Γ ( s + 1 2 ) Γ ( ν + 1 2 - s ) ( 1 - z 2 ) - s ) ( - j ) + j = 0 res s ( ( Γ ( s ) Γ ( 1 - ν 2 - s ) Γ ( ν + 1 2 - s ) ( 1 - z 2 ) - s ) Γ ( s + 1 2 ) ) ( - j - 1 2 ) ) - ν cos ( π ν 2 ) sin ( π ν ) 4 π 3 / 2 ( j = 0 res s ( ( Γ ( s + 1 2 ) Γ ( - s - ν 2 ) Γ ( ν 2 - s ) ( 1 - z 2 ) - s ) Γ ( s ) ) ( - j ) + j = 0 res s ( ( Γ ( s ) Γ ( - s - ν 2 ) Γ ( ν 2 - s ) ( 1 - z 2 ) - s ) Γ ( s + 1 2 ) ) ( - j - 1 2 ) ) /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 Condition ChebyshevT ν z z ν 2 -1 ν 2 3 2 -1 j 0 DirectedInfinity 1 Subscript res s Gamma 1 -1 ν 2 -1 -1 s Gamma s 1 2 Gamma ν 1 2 -1 -1 s 1 -1 z 2 -1 s -1 j j 0 DirectedInfinity 1 Subscript res s Gamma s Gamma 1 -1 ν 2 -1 -1 s Gamma ν 1 2 -1 -1 s 1 -1 z 2 -1 s Gamma s 1 2 -1 j -1 1 2 -1 ν ν 2 -1 ν 4 3 2 -1 j 0 DirectedInfinity 1 Subscript res s Gamma s 1 2 Gamma -1 s -1 ν 2 -1 Gamma ν 2 -1 -1 s 1 -1 z 2 -1 s Gamma s -1 j j 0 DirectedInfinity 1 Subscript res s Gamma s Gamma -1 s -1 ν 2 -1 Gamma ν 2 -1 -1 s 1 -1 z 2 -1 s Gamma s 1 2 -1 j -1 1 2 1 -1 z 2 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["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["Cos", "[", FractionBox[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"]], "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["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"]], "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", " ", RowBox[List["Sin", "[", FractionBox[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["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]], "+", 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"]], "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "s"]], "+", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["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["2", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], "]"]], "<", "1"]]]]]]]]

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

 2007-05-02