Hi, I'm doing a course on PDEs and we have started looking at Fourier series. I am revising and 2 questions have come up that I am stuck on. One wants the Fourier series expansion of the extension of f(x)=x , -pi<x<pi, and the other the same but with f(x)=|x|.
I can start both but then I get stuck. I know that I need to use complex Fourier series and that you start by finding the Fourier coefficient f^(n) (thats f with a hat, not f to the power n) using integration by parts. This is fine. Its when you have to put everything together at the end and for some reason use f^(0) that I start getting confused. If anyone could help me solve these particualr questions or just give me a general step-by-step guide that would be great. This is what I have so far....(im not very good with the inserting of the symbols so bear with me!)
For f(x)=x, -pi<x<pi
f(x) = 1/(2*pi)*sum(from n= - infinity to infinity) f^(n)*e^(i*n*x) and
f^(n) = integral (between -pi and pi) f(x)*e^(-i*n*x)
I can substitute an x in wherever there is an f(x) and then integrate f^(n) by parts to eventually get f^(n) = (2*pi*((-1)^n)*i)/n
From here I am stuck!
Firstly am I right so far? If so, what do I do next?