Let $H$ be some complex Hilbert space. It was shown in many references that for some $0<r<1,$ we have
\begin{equation} A^r = \frac{\sin(\pi r)}{\pi} \int_0^\infty s^{r-1}A(s+A)^{-1}ds, \end{equation} for every normal bounded operator $A\in \mathcal{L}(H) $ with spectrum $$\sigma(A) \subset \Gamma:= \{z\in \mathbb{C}\backslash\{0\},\quad -\pi< Arg(z)<\pi\}.$$ Let now $A_0$ be bounded normal with spectrum $\sigma(A_0) \in \Gamma.$ Can we find $\delta >0$ such that the above formula is still true on $$B(A_0, \delta):= \{A \quad\text{is bounded in}\quad \mathcal{L}(H), \|A_0-A\|\leq \delta\}?$$ It looks like a perturbation result. That's why I tried to find some useful hint here... Thank you for any further hint.