Difference between revisions of "Chebyshev U"
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(Created page with " =Properties= <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem (Orthogonality):</strong> The following formulas hold: $$\int_{-1}^1 \df...") |
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| + | The Chebyshev polynomials of the second kind are orthogonal polynomials defined by | ||
| + | $$U_n(x) = \sin(n \mathrm{arcsin}(x)).$$ | ||
=Properties= | =Properties= | ||
Revision as of 10:35, 23 March 2015
The Chebyshev polynomials of the second kind are orthogonal polynomials defined by $$U_n(x) = \sin(n \mathrm{arcsin}(x)).$$
Properties
Theorem (Orthogonality): The following formulas hold: $$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0 \\ \pi &; m=n=0 \end{array} \right.$$ and $$\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.$$
Proof: █
