= Solution
With the stated <Fourier transform> convention, the exponential transform is a shifted <Dirac delta distribution>:
$$
\boxed{\widetilde f_1(\omega)=2\pi\delta(\omega-\omega_0).}
$$
Writing the sine as $(e^{i\omega_0t}-e^{-i\omega_0t})/(2i)$ then gives
$$
\boxed{\widetilde f_2(\omega)=\frac\pi i[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)].}
$$
These are <tempered distributions>, since the time-domain functions are not integrable on the full real line. The second formula also correctly vanishes when $\omega_0=0$.
Back to article page
