For a causal function,
each term in *U*(*Z*) will be smaller if *Z* is taken to be
inside the circle |*Z*| < 1 rather than on the rim |*Z*|=1.
Thus, convergence at *Z*=0 and on the circle |*Z*|=1 implies
convergence everywhere inside the unit circle.
So boundedness combined with causality means convergence in the unit circle.

Convergence at but not on the circle would refer to a causal function with infinite energy, a case of no practical interest. What function converges on the circle, at , but not at ? What function converges at all three places, , , and ?

10/21/1998