I am trying to find compute the complex Fourier series of the following function:
$$f(t) = t^3$$ $$-\frac32 \le t \le \frac32$$ $$f(t) = f(t+3)$$
I am using the generic function for the complex fourier series and complex fourier coefficient, however I am getting a very complicated integral for the coefficient that doesn't seem right. Could someone give me a hand?
A very complicated integral? It doesn't look so. Integration by parts gives:
$$ \frac{2}{3}\int_{-3/2}^{3/2}t^3 \sin\left(\frac{2\pi}{3}t\right)\,dt = \frac{27\,(-1)^n}{4 n^3 \pi^3}\left(6-n^2 \pi^2\right)\tag{1}$$ hence our Fourier series, since $f(t)$ is an odd function, is given by: $$ \sum_{n\geq 1}\frac{27\,(-1)^n}{4 n^3 \pi^3}\left(6-n^2 \pi^2\right)\sin\left(\frac{2\pi}{3}t\right)\tag{2}$$ and in order to put it in complex form we just need to recall that $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$.