Timeline for The $2\pi$ in the definition of the Fourier transform

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Mar 23, 2017 at 18:20	comment	added	LSpice		Does this really deserve to be called 'elegant'? It can also simplify formulæ dramatically if you don't have to worry about signs, but declaring all signs to be $+1$ and leaving it up to the reader to reconstruct them is surely not a good idea.
Mar 23, 2017 at 15:00	comment	added	nfdc23		The convention to use $e^{2\pi i n x}$ for Fourier series expresses the fact that the self-duality of $\mathbf{R}$ determined by convention 2 identifies the discrete subgroup $\mathbf{Z}$ with its counting measure as Pontryagin dual to the compact quotient $\mathbf{R}/\mathbf{Z}$ with its volume-1 measure (quotient measure from the unique self-dual measure ${\rm{d}}x$ on $\mathbf{R}$).
Mar 23, 2017 at 8:29	comment	added	coudy		Yet Trèves uses convention 2 in his books. I am wondering what explains the success of the $\pi = 1$ trick? Maybe this just amounts to redefining the exponential as $e^{2\pi x}$ in convention 2. At least with Fourier series it is customary to define $e_n(x) = e^{2\pi i n x}$ to alleviate notations.
Mar 23, 2017 at 4:57	history	answered	clyde	CC BY-SA 3.0