A little while ago I happened across a curious formula that blew my mind (no idea what it's called):
$e^{\frac{d}{dx}}f(x)=f(x+1)$
I played around with it a bit and managed to prove it using the Taylor expansion of the exponential and a bit of trickery with the arguments, so I have no issue with it's accuracy.
My question: Is this just convenient notation (to avoid writing out the Taylor series summation) or does it have well defined meaning in and of itself? I tried rearranging and evaluating the derivative inside the exponential for a trial function ($f(x)=x^2$) and it came out with nonsense.
It's much more than convenient notation, but to see what's going on it helps to write it in a more general form as
$$e^{t \frac{d}{dx}} f(x) = f(x + t).$$
This is a version of the exponential map in Lie theory. $\frac{d}{dx}$ can be interpreted as a vector field on $\mathbb{R}$, and $e^{t \frac{d}{dx}}$ can be interpreted as the one-parameter group of diffeomorphisms of $\mathbb{R}$ given by flowing along the vector field. Passing from one to the other is an important operation in Lie theory, differential geometry, and functional analysis, at the very least.