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vpde_lecture20 [2020/03/19 12:41] trinh |
vpde_lecture20 [2020/03/19 15:59] trinh [Fourier series for even and odd- extensions] |
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| === Odd- and even extensions of $f(x) = x^2$ === | === Odd- and even extensions of $f(x) = x^2$ === | ||
| - | We'll draw the odd and even periodic extension of $f(x) | + | We'll draw the odd and even periodic extension of $f(x) = x^2$ originally defined on $[0, \pi]$, and then extended in an even or odd manner to $[-\pi, \pi]$. So for example |
| + | $$ | ||
| + | f_e(x) = \begin{cases} | ||
| + | x^2 & x\in[0, \pi] \\ | ||
| + | x^2 & x\in[-\pi, 0] | ||
| + | \end{cases} | ||
| + | $$ | ||
| + | is the even extension, while | ||
| + | $$ | ||
| + | f_o(x) = \begin{cases} | ||
| + | x^2 & x\in[0, \pi] \\ | ||
| + | -x^2 & x\in[-\pi, 0] | ||
| + | \end{cases} | ||
| + | $$ | ||
| + | is the odd extension. | ||
| - | === Fourier series | + | Once you have drawn (or derived) the extension, then it becomes a simple matter to calculate the Fourier series. For example, since $f_e(x)$ is an even function then only cosines are present, and |
| - | + | $$ | |
| - | We'll then do two examples. One will be the Fourier series for full $2\pi$-periodic extension of $e^x$ defined on $[0, 2\pi]$. The other will be the even extension | + | f_e(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(nx), |
| + | $$ | ||
| + | where | ||
| + | $$ | ||
| + | a_n = \frac{2}{\pi} \int_0^\pi x^2 \cos(nx) \, \mathrm{d}x. | ||
| + | $$ | ||
| + | Above, we have used the even property to double up the integral over the positive $x$ values. Similarly, | ||
| + | $$ | ||
| + | f_o(x) \sim \sum_{n=1}^\infty b_n \sin(nx), | ||
| + | $$ | ||
| + | where | ||
| + | $$ | ||
| + | b_n = \frac{2}{\pi} \int_0^\pi x^2 \sin(nx) \, \mathrm{d}x. | ||
| + | $$ | ||