In some lecture notes on Harmonic Analysis from Terence Tao here, he defines the fourier transform by $$\hat{f}(\tau)=\int_{\mathbb{R}}e^{-2\pi i t\tau}f(t)dt$$ and then says
This is really the best place to put the $2\pi$; listen to the harmonic analysts and representation theorists on this one, and ignore the PDE people, physicists, and engineers!
But why is this the best spot for it? It seems cleaner than having a factor of $\frac{1}{2\pi}$ or $\frac{1}{\sqrt{2\pi}}$ in front of the integral. But it isn't so clean that you forget there is a $2\pi$ at all, like if you hide it in the measure $dt$.
Are there other reasons? Better Reasons? Reasons that only reveal themselves after working in harmonic analysis for a while? Any answers would be appreciated.
There are conventions that are standard in different fields of application. For example, mathematicians like the $1/\sqrt{2 \pi}$ factor because it makes the FT unitary. Physicists like the $1/(2 \pi)$ factor because it helps them remember conservation laws that are deduced from Parseval's Theorem. (I fit into this camp.) And electrical engineers like the $2 \pi$ in the exponential because they like to operate in terms of cycles per second rather than with angular frequencies.
I may be over-generalizing, but as I have experience in all three camps, I think my opinion is grounded in some fact.