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--- math121a-f23:hw_8 [2023/10/20 22:06]
pzhou
+++ math121a-f23:hw_8 [2023/10/24 21:43] (current)
pzhou
@@ Line 5: / Line 5: @@
 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) = a_0 + \sum_{n=1}^\infty a_n \cos(n \pi x) + b_n \sin(n \pi x).
+ 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  $b_n$ ?
@@ Line 12: / Line 12: @@
  f(x) = \begin{cases} 1 & 0 < x < 1/2 \cr
 & 1/2 \leq x \leq 1
-\end{\cases}
+\end{cases}
 find  $a_0, a_1, b_1$  and plot the truncated Fourier series
- $a_0 + a_1 \cos(\pi x) + b_1 \sin(\pi x).$
+ a_0 + a_1 \cos(2 \pi x) + b_1 \sin(2 \pi x).
-How does the resemble your original given function?
+How does this resemble your original given function?
@@ Line 24: / Line 26: @@
 And suppose  $f(0) = 1$ . Can you solve  $f(t)$  for  $t > 0$ ?
-For the following problems, you need to apply Laplace transformation to the equation, and get the  $F(p)$  for the desired  $f(t)$ , and do the inverse Laplace transformation to get back the desired answer.
 . Consider the following equation, for  $t>0$ ,

Lecture Notes