I have a problem involving a textbook problem involving Fourier transforms.
a) prove that a Fourier transform of a function $f(t)$ with period $a$ is zero $(F(\omega)=0)$
I've tried the following $f(t)=e^{2\pi it/a}$
$$F(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{i2\pi t/a}e^{i\omega t}dt=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{i(\omega+2\pi/a)t}dt$$
from which we get: $$\lim_{T\to\infty}\frac{e^{i(\omega+2\pi/a)T}-e^{-i(\omega+2\pi/a)T}}{i(\omega+2\pi/a)}=\lim_{T\to\infty}\frac{a \sin{\frac{\omega a +2\pi}{a}} T}{\omega a+2\pi}$$
but I don't know where to go from here
and the second question (which I'm completely clueless about) is the following:
b) Prove that the Fourier transform of $tf(t)$ is equal to $\frac{1}{i} \frac{dF(\omega)}{d\omega}$ that is: $\mathcal{F}[tf(t)](\omega)=\frac{1}{i} \frac{dF(\omega)}{d\omega}$
can anyone help?
First, if $f$ is periodic and non-zero then $f$ is not integrable, so the Fourier transform is not given by that integral.
Second, the idea that if $f$ is periodic then $\hat f=0$ is just silly. Let $f=1$. Once we figure out what we might mean by $\hat f$, it's easy to see that $\hat f=\delta$, a dirac "delta function".
(In general, any periodic continuous function $f$ is a tempered distribution; the Fourier transform is an isomorphism on the space of tempered distributions, so if $f\ne0$ then $\hat f\ne0$.)