Fourier Series

• Aug 17th 2007, 12:56 AM
atwinix
Fourier Series
Hello,

Can anyone please help me with the attached file. Questions (iv) to (vi). Its related to Fourier Series.
• Aug 17th 2007, 11:02 AM
CaptainBlack
Quote:

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
• Aug 17th 2007, 08:29 PM
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
• Aug 17th 2007, 09:20 PM
CaptainBlack
Quote:

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
• Aug 18th 2007, 01:13 AM
atwinix
Oh!!! So that's the catch! I shall try it.