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vpde_lecture20 [2020/03/19 12:41] trinh |
vpde_lecture20 [2020/03/19 15:50] 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 |
- | + | $$ | |
- | === Fourier series of $f(x) = e^x$ === | + | f_e(x) = \begin{cases} |
- | + | x^2 & x\in[0, \pi] \\ | |
- | We'll then do two examples. One will be the Fourier series for full $2\pi$-periodic extension of $e^x$ defined on $[0, 2\pi]$. The other will be the even extension | + | x^2 & x\in[-\pi, 0] |
+ | \end{cases} | ||
+ | $$ | ||
+ | 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. |