html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ChebyshevT

 http://functions.wolfram.com/07.04.06.0084.01

 Input Form

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

 MathML Form

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

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

 2007-05-02