Differences

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--- vpde_lecture20 [2020/03/19 15:49]
trinh [Fourier series for even and odd- extensions]
+++ vpde_lecture20 [2020/03/19 15:59]
trinh [Fourier series for even and odd- extensions]
@@ Line 42: / Line 42: @@
 === Odd- and even extensions of $f(x) = x^2$ ===
-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 $[-\pi, pi]$.
+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.
+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
+$$
+f_e(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(nx),
+$$
+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, the Fourier series for the odd extension is
+$$
+f_o(x) \sim \sum_{n=1}^\infty b_n \sin(nx),
+$$
+where
+$$
+b_n = \frac{2}{\pi} \int_0^\pi x^2 \sin(nx) \, \mathrm{d}x.
+$$

Trinh @ Bath