Difference between revisions of "Chebyshev U"
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=Properties= | =Properties= | ||
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| + | <strong>Theorem:</strong> The polynomials $T_n(x)$ and $U_n(x)$ are two independent solutions of the following equation, called Chebyshev's equation: | ||
| + | $$(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+n^2y=0.$$ | ||
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| + | <strong>Proof:</strong> █ | ||
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<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
Revision as of 10:36, 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: The polynomials $T_n(x)$ and $U_n(x)$ are two independent solutions of the following equation, called Chebyshev's equation: $$(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+n^2y=0.$$
Proof: █
Theorem: The following formula holds: $$\int_{-1}^1 \dfrac{U_m(x)U_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0\\ 0 &; m=n=0. \end{array} \right.$$
Proof: █
