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authorjoshuab <>2010-06-30 20:24:53 +0000
committerbnewbold <bnewbold@adelie.robocracy.org>2010-06-30 20:24:53 +0000
commit538341232274e1d81e5c6bb04634fa195bfe45a9 (patch)
treef916b96ab29b9d78a293e45f7152186069b3cbbf
parent0f14772c07e7d657c35175330d32598319719d04 (diff)
downloadafterklein-wiki-538341232274e1d81e5c6bb04634fa195bfe45a9.tar.gz
afterklein-wiki-538341232274e1d81e5c6bb04634fa195bfe45a9.zip
trying to fix integrals
-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
-$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2$
+$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$