Integrals of matrix functions

Integrals of matrix functions

I've stumbled across some math I've never really encountered before, and I
would love it if someone could provide me with some useful references and
texts on it. I'm dealing with integration over the space of matrices, as
in Random Matrix Theory and such. Specifically, I think I've narrowed my
confusion down to two different types of matrix integration, some of the
form \begin{equation}G=\int dA \,f(A)\end{equation} Where $A$ is a matrix
of some sort, and $f(A)$ is also a matrix. Then there are integrals of the
form \begin{equation}H=\int dA \,f(|A|)\end{equation} Where $|A|$ is the
determinant. This also includes integrals involving $\mathrm{tr}(A)$,
i.e., integrals of scalar functions of matrices. Essentially, integrals of
matrix functions of matrices, and integrals of scalar functions of
matrices over some measure of matrices.
Also, is there any good literature on extensions of contour integration
and complex analysis to these sorts of matrix functions?
Sorry for the long question.