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C. Results from Analytic Function Theory

C.8.2 Poisson-Jensen Formula for the Half-Plane

Lemma C.1  Consider a function $g(z)$ having the following properties

(i)
$g(z)$ is analytic on the closed RHP;

(ii)
$g(z)$ does not vanish on the imaginary axis;

(iii)
$g(z)$ has zeros in the open RHP, located at $a_1, a_2, \ldots, a_n$;

(iv)
$g(z)$ satisfies $\lim_{\vert z\vert\rightarrow \infty}\frac{\vert\ln g(z)\vert}{\vert z\vert}=0$.

Consider also a point $z_0=x_0+jy_0$ such that $x_0>0$; then


\begin{displaymath} \ln \vert g(z_0)\vert=\sum_{i=1}^n\ln \left \vert \frac{z_0... ...c{x_0}{x_0^2+(\omega-y_0)^2} \ln \vert g(j\omega)\vert)d\omega \end{displaymath} (C.8.13)

Proof

Let


\begin{displaymath}\tilde{g}(z)\stackrel{\rm\triangle}{=}g(z)\prod_{i=1}^n \frac{z+a_i^*}{z-a_i} \end{displaymath} (C.8.14)

Then, $\ln \tilde{g}(z)$ is analytic within the closed unit disk. If we now apply Theorem C.9 to $\ln \tilde{g}(z)$, we obtain


\begin{displaymath}\ln \tilde{g}(z_0)= \ln {g(z_0)}+ \sum_{i=1}^n\ln \left ( \fr... ...}\frac{x_0}{x_0^2+(\omega-y_0)^2}\ln \tilde{g}(j\omega)d\omega \end{displaymath} (C.8.15)

We also recall that, if $x$ is any complex number, then $\Re\{\ln x\}=\Re\{\ln\vert x\vert+j\angle x\}=\ln \vert x\vert$. Thus, the result follows upon equating real parts in the equation above and noting that


\begin{displaymath}\ln \left \vert\tilde{g}(j\omega) \right\vert=\ln \left \vert g(j\omega) \right\vert \end{displaymath} (C.8.16)

$\Box \Box \Box $