Difference between revisions of "Orthogonality of Chebyshev U on (-1,1)"
From specialfunctionswiki
(Created page with "==Theorem== The following formula holds for $m, n \in \{0,1,2,\ldots\}$: $$\int_{-1}^1 \dfrac{U_m(x)U_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfra...") |
(No difference)
|
Revision as of 22:42, 19 December 2017
Theorem
The following formula holds for $m, n \in \{0,1,2,\ldots\}$: $$\int_{-1}^1 \dfrac{U_m(x)U_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0\\ 0 &; m=n=0. \end{array} \right.,$$ where $U_n$ denotes Chebyshev polynomials of the second kind.
