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 at least one of f and 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.
Definitions 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 at least one of f and 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 z 0 {\displaystyle z_{0}} of the complex plane, then there exists an integer n such that
( z − z 0 ) n f ( z ) {\displaystyle (z-z_{0})^{n}f(z)}
is holomorphic and nonzero in a neighbourhood of z 0 {\displaystyle z_{0}} (this is a consequence of the analytic property). If n > 0, then z 0 {\displaystyle z_{0}} is a pole of order (or multiplicity) n of f. If n < 0, then z 0 {\displaystyle z_{0}} is a zero of order | n | {\displaystyle |n|} of f. Simple zero and simple pole are terms used for zeroes and poles of order | n | = 1. {\displaystyle |n|=1.} 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. Because of the order of zeros and poles being defined as a non-negative number n and the symmetry between them, it is often useful to consider a pole of order n as a zero of order −n and a zero of order n as a pole of order −n. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at z = 1. Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(z) = 1/2. In a neighbourhood of a point z 0 , {\displaystyle z_{0},} a nonzero meromorphic function f is the sum of a Laurent series with at most finite principal part (the terms with negative index values):
f ( z ) = ∑ k ≥ − n a k ( z − z 0 ) k , {\displaystyle f(z)=\sum _{k\geq -n}a_{k}(z-z_{0})^{k},}
where n is an integer, and a − n ≠ 0. {\displaystyle a_{-n}\neq 0.} Again, if n > 0 (the sum starts with a − | n | ( z − z 0 ) − | n | {\displaystyle a_{-|n|}(z-z_{0})^{-|n|}} , the principal part has n terms), one has a pole of order n, and if n ≤ 0 (the sum starts with a | n | ( z − z 0 ) | n | {\displaystyle a_{|n|}(z-z_{0})^{|n|}} , there is no principal part), one has a zero of order | n | {\displaystyle |n|} .
… excerpt ends here. Continue reading the full article.
