Contour integral for unbounded operators

Question

For a bounded self-adjoint operator $A$ and a function $f$ analytic in a neighbourhood of $\sigma(A)$, we have $$ f(A) = \frac{1}{2\pi i} \int_\gamma f(z) \, (z-A)^{-1} \, dz, $$ where $\gamma$ is a contour going once around the spectrum of $A$, in counter-clockwise direction. Does something similar hold for unbounded operators? And if yes, how do I have to choose the contour?