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| -rw-r--r-- | Problem Set 1.page | 10 |
1 files changed, 6 insertions, 4 deletions
diff --git a/Problem Set 1.page b/Problem Set 1.page index 885d438..a58e2dc 100644 --- a/Problem Set 1.page +++ b/Problem Set 1.page @@ -14,7 +14,8 @@ Cook up other examples and post them on the wiki! -2. Let $X$ be any set. Show that the cardinality of $2^{X}$ is larger than the cardinality of $X$. Hint: Let $f: X \to 2^X$ be a bijection. Consider the set of all elements $x \in X$ such that $x$ is not an element of $f(x)$. +2. Let $X$ be any set. Show that the cardinality of $2^{X}$ is larger than the cardinality of $X$. +(Hint: Let $f: X \to 2^X$ be a bijection. Consider the set of all elements $x \in X$ such that $x$ is not an element of $f(x)$.) ## Fourier Series @@ -22,13 +23,14 @@ 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. $f(x) = \sin^3(3x)\cos^2(4x)$ - - $g(x) = x(x-2\pi)$ (Hint: Use integration by parts) + - $g(x) = x(x-2\pi)$ + (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 $\sin^4(x)$.) 3. Compute the exponential Fourier coefficients of $\sin^2(x)$: -$a_n = \frac{1}{\sqrt(2\pi)} \int_0^{2\pi} sin^2(x) e^{-inx} dx$ +$a_n = \frac{1}{\sqrt 2\pi} \int_0^{2\pi} \sin^2(x) e^{-inx} dx$ and use this to show that -$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2$ +$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$ |