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## C. Results from Analytic Function Theory

#### C.8.4 Poisson-Jensen Formula for the Unit Disk

Lemma C.2  Consider a function having the following properties:

(i)
is analytic on the closed unit disk;
(ii)
does not vanish on the unit circle;
(iii)
has zeros in the open unit disk, located at .

Consider also a point such that ; then

 (C.8.27)

##### Proof

Let

 (C.8.28)

Then is analytic on the closed unit disk. If we now apply Theorem C.10 to , we obtain

 (C.8.29)

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

 (C.8.30)

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.

If

 (C.8.31)

then

 (C.8.32)

##### Proof

We first note that . We then apply the Poisson-Jensen formula to and at to obtain

 (C.8.33)

We thus have that

 (C.8.34) (C.8.35)

The result follows upon subtracting Equation (C.8.35) from (C.8.34), and noting that

 (C.8.36)

Remark C.3  Further insights can be obtained from Equation (C.8.32) if we assume that, in (C.8.31), and are polynomials;

 (C.8.37) (C.8.38)

then

 (C.8.39)

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

 (C.8.40)

where and are the zeros and the poles of , respectively, that lie outside the unit circle .

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