Zeroes and poles

Rational functions have two important characteristics. For a function $h(z)=\frac{g(z)}{f(z)}$:

• Poles — are the roots of the denominator. These are points where the function fails to be analytic.
• Zeroes — are the roots of the numerator. If $h(z)$ is analytic at $z_0$, and $h(z_0)=0,h^{\prime }(z_0)=0,\dots ,h^{n-1}(z_0)\ne 0,h^m(z_0)\ne 0$, then we say that $h(z)$ has a zero of order $m$ at $z_0$.

We have an additional proposition. If $f(z)=\frac{g(z)}{h(z)}$, with $g(z)$ and $h(z)$ both analytic at $z_0$, and $g(z_0)\ne 0$, and $h(z)$ has a zero of order $m$ at $z_0$, then $f(z)$ has a pole of order $m$ at $z_0$. Then, if $g(z)$ has a zero of order $n$ at $z_0$ and $h(z)$ has a zero of order $m$ at $z_0$, then $f(z)$ has a:

• Pole of order $m-n$ if $m>n$
• Neither pole nor zero if $m=n$
• Zero of order $n-m$ if $n>m$

See also

• Time response, for interpretation on how poles influence the behaviour of systems