 Zeros and poles

Zeros and poles describe the positive and negative local orders of a meromorphic function. Locally $f(z)=(z-a{)}^{m}g(z)$ with $g$ holomorphic and nonzero at $a$: $m>0$ is a zero, $m<0$ is a pole, and $m=0$ is a regular nonzero value. A simple pole has $m=-1$, and the order of vanishing records $m$.

Zeros and poles are fundamental concepts in the field of complex analysis, particularly in control theory and signal processing, where they are used to analyze and design linear systems. ### Zeros: - **Definition**: Zeros are the values of the input variable (often \( s \) in the Laplace domain) that make the transfer function of a system equal to zero.