I am trying to find Exponential Fourier series coefficients for a periodic parabolic function
$$ f(t) = t^2, \quad -\pi \le t \le \pi $$
using derivative property of Fourier series. As otherwise, it's a bit tedious to go and compute FS coefficients by integrating them through formula we have for Fourier series coefficients.
The results I am getting is incorrect (I am differentiating parabolic function thrice to get two impulse trains, FS coefficients for impulse trains are $1/T$ and then applying derivative rule), FS coefficients that we must be getting should be purely real and even but I am getting them as purely imaginary. Can someone please correct me where I am wrong.
Thanks a lot!!