Difference between revisions of "Chebyshev T"
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Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined by | Chebyshev polynomials of the first kind are [[orthogonal polynomials]] defined by | ||
| − | $$T_n(x) = \cos(n \mathrm{arccos}(x))$$ | + | $$T_n(x) = \cos(n \mathrm{arccos}(x)).$$ |
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The polynomials $T_n(x)$ and $U_n(x)$ are two independent solutions of the following equation, called Chebyshev's equation: | ||
| + | $$(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+n^2y=0.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$T_{n+1}(x)-2xT_n(x)+T_{n-1}(x)=0.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Orthogonality):</strong> 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 | ||
| + | $$\int_{-1}^1 \dfrac{U_m(x)U_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} | ||
| + | 0 &; m \neq n \\ | ||
| + | \dfrac{\pi}{2} &; m=n\neq 0\\ | ||
| + | 0 &; m=n=0. | ||
| + | \end{array} \right.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 10:27, 23 March 2015
Chebyshev polynomials of the first kind are orthogonal polynomials defined by $$T_n(x) = \cos(n \mathrm{arccos}(x)).$$
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: $$T_{n+1}(x)-2xT_n(x)+T_{n-1}(x)=0.$$
Proof: █
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: █
