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Let $u$ be a function on a domain $\Omega$ with some fixed boundary condition.
I have recently seen a notation $e^{\tau \Delta}u$ as meaning the the time evolution of $u$ by diffusion for a time $\tau$. I'm curious where this notation comes from, and more generally, what is meant by a function of a differential operator.
There is an entire branch of mathematics dedicated to the question what is $e^{tA}$ for a differential operator $A$: One parameter Semigroup theory and theory of evolution equations. You could take a look at the Short_Course from Engel and Nagel or directly consult PDE literature (like Chapter 7.4 in Evans' book on PDEs).
If you're asking about applying more general functions to operators, then you are finding yourself in the field of functional calculus. If you have a favorite book on functional analysis at hand, you could try to look this term up, or you could start from wikipedia.
