Solving probability transform functional equations for numerical inversion

Many methods for numerically inverting transforms require values of the transform at complex arguments. However, in some applications, the transforms are only characterized implicitly via functional equations. This is illustrated by the busy-period distribution in the M/G/1 queue. In this paper the authors provide conditions for iterative methods to converge for complex arguments. Moreover, they show that stochastic monotonicity properties can provide useful bounds.