Sine and cosine series

Odd and even symmetry kills half the coefficients, so a function given on a half interval can be extended to get either a pure sine or a pure cosine expansion.

Notes on Diffy Qs: Differential Equations for Engineers, Sine and cosine series

Status

FlagsLoading your progress…

5 exercises

Exercise 101

Let $f(t) = \nicefrac{t}{3}$ on $0 \leq t < 3$ .

Find the Fourier series of the even periodic extension.
Find the Fourier series of the odd periodic extension.

2.5

Exercise 102

Let $f(t) = \cos(2t)$ on $0 \leq t < \pi$ .

Find the Fourier series of the even periodic extension.
Find the Fourier series of the odd periodic extension.

2.5

Exercise 103

Let $f(t)$ be defined on $0 \leq t < 1$ . Consider the average of the two extensions $g(t) = \frac{F_{\text{odd}}(t)+ F_{\text{even}}(t)}{2}$ .

What is $g(t)$ if $0 \leq t < 1$ (Justify!)
What is $g(t)$ if $-1 < t < 0$ (Justify!)

2.5

Exercise 104

Let $f(t) = \sum_{n=1}^\infty \frac{1}{n^2} \sin(nt)$ . Solve $x''- x = f(t)$ for the Dirichlet conditions $x(0) = 0$ and $x(\pi) = 0$ .

2.5

Exercise 105

Let $f(t) = t + \sum_{n=1}^\infty \frac{1}{2^n} \sin(nt)$ . Solve $x'' + \pi x = f(t)$ for the Dirichlet conditions $x(0) = 0$ and $x(\pi) = 1$ . Hint: Note that $\frac{t}{\pi}$ satisfies the given Dirichlet conditions.

4.0