-Integrate[ChebyshevT[n, x]^2 Log[ChebyshevT[n, x]^2] (1/Sqrt[1 - x^2]), {x, -1, 1}] == (-Pi) (1 - KroneckerDelta[n, 0]) (Log[2] - 1) /; Element[n, Integers] && n >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[SuperscriptBox[RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", "x"]], "]"]], "2"], " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", "x"]], "]"]], "2"], "]"]], " ", FractionBox["1", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], RowBox[List["\[DifferentialD]", "x"]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", "\[Pi]"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", " ", "0"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", "1"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", " ", "Integers"]], "]"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <msup> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> x </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#948; </mi> <mrow> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <apply> <power /> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#948; </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["-", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["ChebyshevT", "[", RowBox[List["n_", ",", "x_"]], "]"]], "2"], " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["ChebyshevT", "[", RowBox[List["n_", ",", "x_"]], "]"]], "2"], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x_", "2"]]]]], RowBox[List["\[DifferentialD]", "x_"]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "0"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", "1"]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

2001-10-29