Meromorphic generalized inverses of operator functions

In this article we give a survey of results on the inversion of holomorphic operator functions under certain conditions and also include a few new results. We study [semi]-Fredholm or relatively regular operators which depend holomorphically on one or several complex variables. We construct [one-sided or generalized] finite-meromorphic inverses of such functions and investigate the global structure of these inverses. Moreover, under suitable assumptions we obtain meromorphic solutions to vector function equations and operator function equations for semi-Fredholm functions which need not be relatively regular.