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Concept
Zeros and poles
Summary
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic (i.e. complex differentiable) in some neighbourhood of z0.
A function f is meromorphic in an open set U if for every point z of U there is a neighborhood of z in which either f or 1/f is holomorphic.
If f is meromorphic in U, then a zero of f is a pole of 1/f, and a pole of f is a zero of 1/f. This induces a duality between zeros and poles, that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros.
A function of a complex variable z is holomorphic in an open domain U if it is differentiable with respect to z at every point of U. Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of U, and converges to the function in some neighbourhood of the point. A function is meromorphic in U if every point of U has a neighbourhood such that either f or 1/f is holomorphic in it.
A zero of a meromorphic function f is a complex number z such that f(z) = 0. A pole of f is a zero of 1/f.
If f is a function that is meromorphic in a neighbourhood of a point of the complex plane, then there exists an integer n such that
is holomorphic and nonzero in a neighbourhood of (this is a consequence of the analytic property).
If n > 0, then is a pole of order (or multiplicity) n of f. If n < 0, then is a zero of order of f. Simple zero and simple pole are terms used for zeroes and poles of order Degree is sometimes used synonymously to order.
This characterization of zeros and poles implies that zeros and poles are isolated, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole.
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