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| -rw-r--r-- | Problem Set 1.page | 10 |
1 files changed, 5 insertions, 5 deletions
diff --git a/Problem Set 1.page b/Problem Set 1.page index 88ec240..d77e2eb 100644 --- a/Problem Set 1.page +++ b/Problem Set 1.page @@ -1,5 +1,5 @@ ## Countability - +$\int f(x)$ 1. Group the following sets according to their cardinality: a. $\mathbb{N} = \{ 1,2,3,4,\dots \}$ @@ -25,10 +25,10 @@ Cook up other examples and post them on the wiki! - $g(x) = x(x-2\pi)$ (Hint: Use integration by parts) 2. Show that -$\int_0^{2\pi} sin^4(x) dx = \frac{3 \pi}{4} $ -(Hint: write out the exponential fourier expansion of $sin^4(x)$.) +$\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 $sin^2(x)$: +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 $ +$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2 $ |