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 Nicolaescu a different strategy is used).