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--- math121a-f23:hw_8 [2023/10/21 04:59]
pzhou
+++ math121a-f23:hw_8 [2026/02/21 14:41] (current)
@@ Line 1: / Line 1: @@
 ====== Homework 8 ======
-. Sine wave decomposition.
+. Sine and Cosine decomposition.
-Suppose you are given a function on an interval, $f(x): [0, 1] \to \R$, such that $f(x)$ vanishes on both end points $f(0)=f(1)=0$. Such function $f(x)$ can be expressed as a sum of 'sine waves'.
+Suppose you are given a function on an interval, $f(x): [0, 1] \to \R$. Such function $f(x)$ can be expressed as a sum of 'sine waves' and cosine waves and constant
-$$ f(x) = \sum_{n=1}^\infty c_n \sin(n \pi x). $$
-Can you figure out a way to determine the coefficients $c_n$?
+$$ f(x) = a_0 + \sum_{n=1}^\infty a_n \cos(2n \pi x) + b_n \sin(2n \pi x). $$
+Can you figure out a way to determine the coefficients $a_n$ and $b_n$?
+Test out your method for the following function
+$$ f(x) = \begin{cases} 1 & 0 < x < 1/2 \cr
+& 1/2 \leq x \leq 1
+\end{cases}
+$$
+find $a_0, a_1, b_1$ and plot the truncated Fourier series
+$$ a_0 + a_1 \cos(2 \pi x) + b_1 \sin(2 \pi x). $$
+How does this resemble your original given function?
@@ Line 12: / Line 25: @@
 $$ f'(t) +  f(t) = 0 $$
 And suppose $f(0) = 1$. Can you solve $f(t)$ for $t > 0$?
 . Consider the following equation, for $t>0$,
@@ Line 18: / Line 32: @@
 . Consider the following equation, for $t>0$,
-$$ (d/dt + 1) (d/dt + 1) f(t) = 0 $$
+$$ [(d/dt)^2 + 1] f(t) = 0 $$
 And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?
-. Consider the following equation, for $t>0$,
+ (bonus, optional). Consider the following equation, for $t>0$,
-$$ [(d/dt)^2 + 1] f(t) = 0 $$
+$$ (d/dt + 1) (d/dt + 1) f(t) = 0 $$
 And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?

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