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C. Results from Analytic Function Theory

C.9 Application of the Poisson-Jensen Formula to Certain Rational Functions

Consider the biproper rational function $\bar{h}(z)$ given by


\begin{displaymath}\bar{h}(z)=z^{\bar{\lambda}}\frac{\bar{f}(z)}{\bar{g}(z)} \end{displaymath} (C.9.1)

$\bar{\lambda}$ is a integer number, and $ \bar{f}(z)$ and $\bar{g}(z)$ are polynomials of degrees $m_f$ and $m_g$, respectively. Then, due to the biproperness of $\bar{h}(z)$, we have that $\bar{\lambda}+m_f=m_g$.

Further assume that

(i)
$\bar{g}(z)$ has no zeros outside the open unit disk,
(ii)
$ \bar{f}(z)$ does not vanish on the unit circle, and
(iii)
$ \bar{f}(z)$ vanishes outside the unit disk at $\beta_1, \beta_2, \ldots, \beta_{m}$.

Define


\begin{displaymath} h(z)=\frac{f(z)}{g(z)} \stackrel{\rm\triangle}{=}\bar{h}\left(\frac{1}{z}\right) \end{displaymath} (C.9.2)

where $f(z)$ and $g(z)$ are polynomials.

Then it follows that

(i)
$g(z)$ has no zeros in the closed unit disk;
(ii)
$f(z)$ does not vanish on the unit circle;
(iii)
$f(z)$ vanishes in the open unit disk at $\bar{\beta}_1, \bar{\beta}_2, \ldots, \bar{\beta}_{m}$, where $\bar{\beta}_i=\beta_i^{-1} $ for $i=1,2,\ldots, \bar{\beta}_{m}$;
(iv)
$h(z)$ is analytic in the closed unit disk;
(v)
$h(z)$ does not vanish on the unit circle;
(vi)
$h(z)$ has zeros in the open unit disk, located at $\bar{\beta}_1, \bar{\beta}_2, \ldots, \bar{\beta}_{m}$.

We then have the following result

Lemma C.3  Consider the function $h(z)$ defined in (C.9.2) and a point $z_0=re^{j\theta}$ such that $r<1$; then


\begin{displaymath} \ln \vert h(z_0)\vert=\sum_{i=1}^{\bar{m}}\ln \left \vert ... ...pi}P_{1,r}(\theta-\omega)\ln \vert h(e^{j\omega})\vert d\omega \end{displaymath} (C.9.3)

where $P_{1,r}$ is the Poisson kernel defined in (C.8.18).

Proof

This follows from a straightforward application of Lemma C.2.

$\Box \Box \Box $