[Math] Fourier-Laplace Transform of Heaviside Step function multiplied to Sine

Question

In a Advanced Solid State lecture I encountered the following assertion-
Fourier Transform of $\Theta(t)\sin(\omega_0 t)$ is $\frac{1}{\omega+\omega_0}-\frac{1}{\omega-\omega_0}+i\pi\delta(\omega+\omega_0)$. I do not see how that is. I know that some kind of regularization by doing the Fourier Transformation in complex integral is involved. But I do not know to go past something of the form
$\lim_{\alpha\rightarrow 0}\int dt \exp(i(\omega+\omega_0)t)\exp(-(\omega+\omega_0)\alpha)$
Interestingly, the wolframalpha output seems to outdo me in this!

Answer 1

You need a table of correspondences for distributions. I use angular frequency $\omega$ and the non-unitary version of the Fourier transform:

$$\sin\omega_0t\Longleftrightarrow -i\pi[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]\\ \Theta(t)\Longleftrightarrow\frac{1}{i\omega}+\pi\delta(\omega)$$

With these correspondences you get

$$\Theta(t)\sin\omega_0t\Longleftrightarrow\frac{1}{2\pi}\mathcal{F}(\Theta(t))* \mathcal{F}(\sin\omega_0t)$$

where $*$ stands for convolution. The result of the convolution is

$$\Theta(t)\sin\omega_0t\Longleftrightarrow -\frac12\left[\frac{1}{\omega-\omega_0}-\frac{1}{\omega+\omega_0}\right]-\frac{i\pi}{2}\left[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)\right]$$

Note that WolframAlpha uses the unitary version of the Fourier transform which gives you different constants.