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http://functions.wolfram.com/07.13.07.0001.01
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GegenbauerC[n, x] == (-(2/(Pi n)))
Integrate[(Sqrt[1 - t^2] ChebyshevU[n - 1, t])/(t - x), {t, -1, 1},
PrincipalValue -> True] /; Element[n, Integers] && n > 0 && -1 < x < 1
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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> <mi> C </mi> <mi> n </mi> <mrow> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 2 </mn> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mi> 𝒫 </mi> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> t </mi> <mo> - </mo> <mi> x </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> x </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> GegenbauerC </ci> <ci> n </ci> <cn type='integer'> 0 </cn> <ci> x </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <pi /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> 𝒫 </ci> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <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> <apply> <ci> ChebyshevU </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> t </ci> </apply> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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