Difference between revisions of "Orthogonality of Chebyshev T on (-1,1)"
From specialfunctionswiki
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==Proof== | ==Proof== | ||
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[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Revision as of 22:30, 19 December 2017
Theorem
The following formula holds for $m,n \in \{0,1,2,\ldots\}$: $$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} \mathrm{d}x = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0 \\ \pi &; m=n=0, \end{array} \right.$$ where $T_m$ denotes Chebyshev polynomials of the first kind.
