I asked the question Fourier series is to Fourier transform what Laurent series is to …? over at MSE, since that's where my questions usually belong to. But since I couldn't find any resources on it, I was wondering if it were actually adequate for migration to MO?

To summarize that question, it is about the similarity between Fourier analysis and the Laurent series on a circle and whether the $r\to\infty$-limit of the latter a) makes sense b) also yields an integration similar to the Fourier series turning into the Fourier transform and c) whether its useful.

no. 1. You've only asked the question a few hours ago. Wait a bit to see what responses you get on MSE. 2. There are some computational mistakes in your post. (See my comment on MSE.) You should first fix that. $\endgroup$