The region of convergence for the z-transform exists in the real and imaginary plane.

The ROC of the z-transform is a ring in the real and imaginary plane centered around the origin.

The range of convergence is dependent on the value or r and not w. The ROC can extend in to zero,

and it can extend outwardly to infinity. The ROC does not contain any poles.

If x[n] is a right-sided sequence, and r(0)=|z| then all values of z will be > r(0). If x[n] is a left-sided

sequence then all values of z will be 0<z<r(0). If x[n] is two sided then ROC will be a ring.

If the z-transform is rational then ROC is bounded by poles or extends to infinity. If x[n] is rational

and right sided then the ROC is outside of the poles. Furthermore if x[n] is causal then ROC includes infinity.

Vice a versa if the x[n] is rational and left sided then the ROC is the innermost part of the plane bounded

by the poles. If x[n[ is not causal then the ROC includes 0.