Complex analysis for matrix functions? Does there exist complex analysis for matrix functions? For example if we have $n\times n$ matrices with elements in $\mathbb C$, and we define functions as usual power series expansions. Does it make sense to start defining some kind "path integrals" or "contour integrals" for them? If so, in what sense?

Clarification:
This question regards functions $\lambda \to A'(\lambda), \lambda \in \mathbb C, A'\in \mathbb C^{n\times n}$
 A: If $A(\lambda)$ is a matrix function of a complex parameter $\lambda$, then you can define holomorphic functions much as you do for scalar functions:
$$
           A'(\lambda)=\lim_{\beta\rightarrow\lambda}  \frac{1}{\beta-\lambda}(A(\beta)-A(\lambda)).
$$
If $A$ is holomorphic in an open region $\Omega$, then $A$ has a (local) Cauchy integral representation, and a local power series expansion in the domain. Such a matrix function is holomorpic iff the elements of the matrix are holomorphic. So you can reduce everything to components. An integral of the matrix function can be reduced to an matrix of the integrals of the components:
$$
    A(\lambda)=\left[ a_{j,k}(\lambda) \right]_{j,k},\\
     \int_{C} A(\lambda)d\lambda=\left[\int_{C}a_{j,k}(\lambda)d\lambda\right]_{j,k}
$$
Cauchy's integral represention holds
$$
                 A(\mu) = \frac{1}{2\pi}\oint_{C}\frac{1}{\lambda-\mu}A(\lambda)d\lambda.
$$
For a constant matrix $A$, you get nice things like
$$
         p(A) = \frac{1}{2\pi i}\oint_{C} p(\lambda)(\lambda I-A)^{-1}d\lambda,
$$
where $p$ is a polynomial. This generalizes to holomorphic functions on a domain that includes the spectrum of $A$. For any such $f,g$, you get $(fg)(A)=f(A)g(A)$. If the order of the poles of $(\lambda I-A)^{-1}$ at the eigenvalues $\lambda_1,\cdots,\lambda_n$ are $r_1,\cdots,r_n$, then you have the minimal polynomial $m(\lambda)=(\lambda-\lambda_1)^{r_1}\cdots(\lambda-\lambda_n)^{r_n}$--it it easy to see that using the above definition gives $m(A)=0$.
