The evaluation of the two-dimensional (2-D) complex integral used in the calculation of the z-transform of a 2-D filter's subsampled impulse response, is always carried out in a particular manner. Traditionally, this approach obviates the problems associated with a 2-D polynomial not satisfying the fundamental theorem of algebra. However, this paper will show that in the particular case of a first-order filter, it is possible to evaluate the complex integral in a different, albeit more complicated way. This entails close inspection of the region of convergence of a 2-D z-transform, and gives an interesting insight into the problems of 2-D complex integrals. Finally, some relevant comments are made on 2-D limit cycles and polyphase structures.