1. ## Fourier Series

Hello,

Can anyone please help me with the attached file. Questions (iv) to (vi). Its related to Fourier Series.

2. Originally Posted by atwinix
Hello,

Can anyone please help me with the attached file. Questions (iv) to (vi). Its related to Fourier Series.
What is the problem with these? The third line in each tells you that the function is periodic with period
$\displaystyle 2 \pi$ so any interval of that length is what you use in the integrations and the basis functions are
$\displaystyle \sin(nx)$ and $\displaystyle \cos(nx)$.

RonL

3. Yes, I know how the Fourier series expansion work.

I am having problem with the integration between the limits for question 6 (iv) as you get something like:

∫ (π+t) cos (nt) dt + ∫ (π-t) cos (nt) dt

4. Originally Posted by atwinix
Yes, I know how the Fourier series expansion work.

I am having problem with the integration between the limits for question 6 (iv) as you get something like:

∫ (π+t) cos (nt) dt + ∫ (π-t) cos (nt) dt
Using integration by parts with $\displaystyle u=(\pi-t)$ and $\displaystyle dv=\cos(nt)$:

$\displaystyle \int_0^\pi (\pi-t) \cos(nt)~dt = \left[ \frac{(\pi-t)}{n} \sin(nt) \right]_0^{\pi}+\int_0^{\pi}\frac{1}{n^2} \sin(nt)~dt$$\displaystyle =\frac{1}{n^2}(1-(-1)^n)$

which is $\displaystyle 2/n^2,\ n=1, 3, 5, ..$ and $\displaystyle 0,\ n=2,4,..$

RonL

5. Oh!!! So that's the catch! I shall try it.