Poleswithin

Poleswithin is a term sometimes used in complex analysis to denote the set of poles of a meromorphic function that lie inside a given region, typically bounded by a simple closed contour. The concept centers on identifying which singularities of the function are contained within the interior of the chosen contour.

Definition and counting. Let f be meromorphic on an open set containing a simple closed contour C. The poleswithin of f with respect to C is the set of points z0 in the interior of C where f has a pole. Each pole z0 has an order (the multiplicity of the pole), so the poleswithin can be counted with multiplicity as the sum of these orders.

Relation to contour integration. The poleswithin feature prominently in the residue theorem: the contour integral of f around C equals 2πi times the sum of residues at all poleswithin, each counted with its multiplicity. This makes the poleswithin essential for evaluating many complex integrals.

Identification methods. To determine the poleswithin, one can locate singularities of f and determine their orders, or apply tools such as the argument principle and Rouché’s theorem to constrain their locations. In practice, one often also uses Laurent expansions around suspected poles to confirm their nature and order.

Examples and applications. For f(z) = (z^2 + 1)/(z(z − 2)), the poles are at z = 0 and z = 2. If C is the unit circle, the poleswithin are {0} (multiplicity 1). The concept is used in evaluating integrals, stability analysis in control theory, and signal processing, where the distribution of poles within a region determines behavior.

Notes. The exact term “poleswithin” is not universally standardized; many texts refer to “poles inside the contour C” or “poles inside the region.”