In my PDE course we have a homework question stating the following:
Let ϑ(x) = x in the interval [-pi, pi ]. Find its Fourier Coefficients.
From my notes on this type of question:
a_o = 2c_o = 1/pi * integral from -pi to pi [f(x) dx]
a_n = c_n + c_(-n) = 1/pi * integral from -pi to pi [f(x) cos(n*x) dx ]
b_n = i(c_n - c_(-n)) = 1/pi * integral from -pi to pi [f(x) sin(n*x) dx]
The Attempt at a Solution
Is it as simple as just a plug and chug based off my noes?
a_o's integration with f(x) = x just is x^2/(2*pi) from -pi to pi so we have
a_o = pi/2 - pi/2 = 0
a_n's integration is just equal to 0 as well.
b_n is just -2(-1)^n/n
So thus, the fourier coefficients here are b_n = [(-2)(-1)^n]/n
for n ≥ 1
Am I understanding the question properly?