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-rw-r--r--Problem Set 1.page10
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
-$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2$
+$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$