$t\in[0,1]$ is given by the norm limit of Riemann sums of the form $$\sum_{j=1}^{\nu}
g(z(\theta_j))[z(t_j)-z(t_{j-1})],\qquad t_{j-1}\leq\theta_j\leq t_j,$$ where $0=t_0 < t_1 < \ldots < t_{\nu -1} < t_{\nu} = 1$.

Denote by $\mathcal{H}(a)$ the algebra of analytic functions whose domain contains an open neighborhood of $\sigma(a)$. The following well-known result is the basis for the [*holomorphic functional calculus*]{}; see, e.g., [@KRI page 206].

The mapping $\mathcal{H}(a)\longrightarrow\mathcal{A}_0$ defined by $f\mapsto f(a)$ is an algebra homomorphism, which maps the constant function $1$ to $I\in\mathcal{A}_0$ and maps the identity function to $a$. If $f(z)=\sum_{j=0}^{\infty}c_jz^j$ is the power series representation of $f\in\mathcal{H}(a)$ over an open neighborhood of $\sigma(a)$, then we have $$f(a)=\sum_{j=0}^{\infty}c_ja^j.$$

Moreover, the following version of the *spectral theorem* holds: $$\sigma(f(a))=f(\sigma(a)).\label{eq_spectral_thm}$$

If $a$ is normal, the following properties also hold:

-   $\|f(a)\|=\|f\|_{\infty,\sigma(a)} := \max_{\lambda \in \sigma (a)} |f(\lambda)|$;

-   $\overline{f(a)}=[f(a)]^*$; in particular, if $a$ is Hermitian then $f(a)$ is also Hermitian if and only if $f(\sigma(a))\subset\mathbb{R}$;

-   $f(a)$ is normal;

-   $f(a)b=bf(a)$ whenever $b\in\mathcal{A}_0$ and $ab=ba$.

Obviously, these definitions and results apply in the case where $a$ is a matrix $A$ with entries in a $C^*$-algebra $\mathcal{A}_0$. In particular, if $f(z)$ is analytic on a neighborhood of $\sigma(A)$, we define $f(A)$ via $$f(A) 
