Fourier transform of a periodic signal

Question

I usually know that we use the continuous Fourier transform on non-periodic signals and Fourier series on periodic signals. However there're signals like $\cos$, $e^{i2\pi ft}$ and others that there is a Fourier transform. Does that mean there is a Fourier transform for periodic signals? If yes, how to calculate it?

Answer 1

If $f$ has a Fourier series representation, then $$\mathcal F[f] = \mathcal F {\left[ \sum_{k \in \mathbb Z} c_k e^{2 \pi i k t/T} \right]} = \sum_{k \in \mathbb Z} c_k \delta {\left( p + \frac {2 \pi k} T \right)} = \frac 1 T \int_0^T f(t) e^{i p t} dt \sum_{k \in \mathbb Z} \delta {\left( p + \frac {2 \pi k} T \right)}.$$