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http://functions.wolfram.com/07.04.06.0086.01
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ChebyshevT[\[Nu], z] == (-((\[Nu] Sin[2 Pi \[Nu]])/(2 Pi^(3/2))))
(Sum[Residue[((Gamma[1/2 + s] Gamma[-s - \[Nu]] Gamma[-s + \[Nu]])/
((1 + z)/2)^s) Gamma[s], {s, -j}], {j, 0, Infinity}] +
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}]) /; Abs[1 + z] < 2
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]], " ", 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", "[", 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[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[RowBox[List["Abs", "[", RowBox[List["1", "+", "z"]], "]"]], "<", "2"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> T </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 2 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ChebyshevT </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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["\[Nu]", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "\[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", "[", 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[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["2", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "+", "z"]], "]"]], "<", "2"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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