I recently posted in Electrical Engineering Stack and no body could give me an answer so I have come to the mathematicians. So in steady state AC circuit analysis we usually transform everything into the frequency domain by using the Fourier transform. For example, if I wanted the frequency domain representation of the voltage across an inductor, I would take the Fourier transform of the differential equation describing it.
$v(t)=L\ di(t)/dt$
$\mathcal{F}[v(t)=L\ di(t)/dt] \rightarrow V(\omega)=j\omega L\ I(\omega)$ where $j$ is the imaginary number.
So when you transform a AC circuit into the frequency domain you also transform the source.
$S(t) = A\ cos(\omega_0t+\phi)$
$\mathcal{F}[S(t)] \rightarrow A\ e^{j\phi}$
So I have not been able to find any proof that the Fourier transform of "Source" $S = A\ e^{j\phi}$. When I attempt to derive it by taking the Fourier transform of "Source" I end up with:
$\mathcal{F}[S] \rightarrow A\ \pi [e^{j\phi}\delta(\omega-\omega_0)+e^{-j\phi}\delta(\omega+\omega_0)]$
I next use the condition that $\omega = \omega_0$ since the frequency will be constant in the system. This leaves me with:
$A\ \pi e^{j\phi}$
So I have a factor of $\pi$ that I do not know what to do with. I am not sure how we can simply say:
$\mathcal{F}[S] \rightarrow A\ e^{j\phi}$
When that is not what you get. Is their any obvious things I am missing? The answer I get looks very close so I feel like I am on the right track.
Thank you!