C. Results from Analytic Function Theory
Consider also a point such that ; then
Then is analytic on the closed unit disk. If we now apply Theorem C.10 to , we obtain
We also recall that, if is any complex number, then . Thus the result follows upon equating real parts in the equation above and noting that
Theorem C.11 (Jensen's formula for the unit disk) Let and be analytic functions on the unit disk. Assume that the zeros of and on the unit disk are and respectively, where none of these zeros lie on the unit circle.
We first note that . We then apply the Poisson-Jensen formula to and at to obtain
We thus have that
Thus, and are all the zeros and all the poles of , respectively, that have nonzero magnitude.
This allows Equation (C.8.32) to be rewritten as
where and are the zeros and the poles of , respectively, that lie outside the unit circle .