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http://functions.wolfram.com/07.04.21.0009.01
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Integrate[(ChebyshevT[m, t] ChebyshevT[n, t])/Sqrt[1 - t^2], {t, -1, 1}] ==
(1/2) Pi KroneckerDelta[m, n] /; Element[m, Integers] && m >= 0 &&
Element[n, Integers] && n >= 0 && m^2 + n^2 != 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[FractionBox[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["m", ",", "t"]], "]"]], RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", "t"]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["t", "2"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "n"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List[SuperscriptBox["m", "2"], "+", SuperscriptBox["n", "2"]]], "\[NotEqual]", "0"]]]]]]]]
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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> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mfrac> <mrow> <mrow> <msub> <mi> T </mi> <mi> m </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> <mo> ≠ </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <ci> ChebyshevT </ci> <ci> m </ci> <ci> t </ci> </apply> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <ci> t </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </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> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <neq /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[FractionBox[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["m_", ",", "t_"]], "]"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List["n_", ",", "t_"]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["t_", "2"]]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "n"]], "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List[SuperscriptBox["m", "2"], "+", SuperscriptBox["n", "2"]]], "\[NotEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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