# Thread: Factoring a complex function

1. ## Factoring a complex function

How can I write the function $f(z)=1-0.7z-0.3z^{-1}$, $z$ complex, as $f^+(z) f^-(z)$, where

1. $f^+(z)$ is free of zeros and singularities **outside and on** the unit circle
2. $f^-(z)$ is free of zeros and singularities **inside and on** the unit circle.

As can be verified, $f$ has a single pole at $z=0$ and two zeros at $z=1$ and $z=3/7$, so I cannot simply take $f^+(z)=f(z)$ and $f^-(z)=1$.

2. ## Re: Factoring a complex function

$\displaystyle f(1)=f^+(1)f^-(1)=0$

implies

$\displaystyle f^+(1)=0$ or $\displaystyle f^-(1)=0$