I was reading through the Functional Analysis by Lax and stumbled onto a proof that I didn't quite understand. More concretely it is abbout the application of Cauchy's integral formula in the proof of Gelfands spectral radius formula.
Lax argues, that by Cauchy's integral formula $$\frac{1}{2\pi i}\int_C\frac{\lambda^n}{\lambda-M}d\lambda=M^n,$$ where $C$ is a path in the resolvent set of $M$ that winds once around $\sigma(M)$, and $$(\lambda-M)^{-1}=\sum_{i=0}^\infty M^i\lambda^{-i-1}$$ holds for any $|\lambda|>\|M\|$. What I don't understand is, that to apply Cauchy's integral formula one needs a holomorphic function on a simply connected set and the path needs to be inside that set. However, $\frac{\lambda^{n-1}}{\lambda-M}$ is only holomorphic on $\rho(M)$ but not on $\sigma(M)$, which is surely inside $\rho(M)$. So how is it valid to argue with Cauchy's integral formula here?