Another reference is
Dimassi-Sjöstrand: Spectral asymptotics in the semiclassical limit, Cambridge University Press (1999)
See in particular Chapter 8 ("Functional Calculus for pseudodifferentiql operators").
The results are in the semiclassical setting. The result which may interest you is Theroem 8.7 which states that if $P$ is a selfsadjoint h-pseudodiff operator and $f\in C^\infty_0$, then f(P) is again a pseudo with leading symbol $f(p((x,\xi))$ , where $p(x,\xi)$ is the leading symbol of $P$.
The method to prove this is based here on a standard Cauchy formula. (I think in the book of Taylor, cited by Livio Liviu Nicolaescu a different strategy is used).
