polecount

Polecount is a term used to describe the number of poles associated with a function or system, typically counted with multiplicity within a specified domain. It is a measure of how many distinct singular dynamics or resonant modes the object exhibits.

In complex analysis, a pole is an isolated singularity where a function behaves like a finite principal part (z−z0)−m multiplied by a analytic function, with m a positive integer called the order of the pole. The pole count within a region is the sum of these orders for all poles located there. This concept underpins tools such as the residue theorem, which relates pole data to integral values.

For rational functions, poles arise at the zeros of the denominator that are not canceled by the numerator. If f(z) = P(z)/Q(z), the poles occur at the zeros of Q that are not common with P, with their multiplicities determined by Q and any cancellations. The pole count is the total number of poles, counting multiplicities. In algebraic contexts, the pole structure can be described by a divisor on a Riemann surface, where the degree of the pole divisor equals the total pole multiplicity.

In applied settings such as control theory and signal processing, polecount often refers to the number of finite poles of a transfer function, which corresponds to the system’s order. The pole locations influence stability and transient response: for continuous-time systems, stability requires poles in the left half-plane; for discrete-time systems, within the unit circle.

Example: G(s) = 1/(s(s+1)) has two simple poles at s = 0 and s = −1.