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@@ -23,18 +23,18 @@ Cook up other examples and post them on the wiki!
1. Compute the Fourier Series of the following functions. Do both the exponential and sin/cos expansions.
- a. $f(x) = \sin^3(3x)\cos^2(4x)$
+ a. f$f(x) = \sin^3(3x)\cos^2(4x)$
- $g(x) = x(x-2\pi)$
(Hint: Use integration by parts)
2. Show that
-$\int_0^{2\pi} \sin^4(x) dx = \frac{3 \pi}{4}$
+$$\int_0^{2\pi} \sin^4(x) dx = \frac{3 \pi}{4}$$
(Hint: write out the exponential fourier expansion of $\sin^4(x)$.)
3. Compute the exponential Fourier coefficients of $\sin^2(x)$:
-$a_n = \frac{1}{\sqrt 2\pi} \int_0^{2\pi} \sin^2(x) e^{-inx} dx$
-and use this to show that
-$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$
+$$a_n = \frac{1}{\sqrt 2\pi} \int_0^{2\pi} \sin^2(x) e^{-inx} dx$$
+and use this to verify that
+$$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$$
# Solutions