Integration of Grassmann variables over invariant functions on flat superspaces

We study integration over functions on superspaces. These functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables. We get a compact expression for the differential operator with respect to the commuting variables which results from Berezin integration over all Grassmann variables. Also, we derive Cauchy-like integral theorems for invariant functions on supervectors and symmetric supermatrices. This extends theorems partly derived by other authors. As a physical application, we calculate the generating function of the one-point correlation function in random matrix theory. Furthermore, we give another derivation of supermatrix Bessel functions for U(k1/k2).