Zeros and poles

In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable.

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.

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 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.

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

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.

a nonzero meromorphic function f is the sum of a Laurent series with at most finite principal part (the terms with negative index values): where n is an integer, and

, the principal part has n terms), one has a pole of order n, and if n ≤ 0 (the sum starts with

is meromorphic at infinity if it is meromorphic in some neighbourhood of infinity (that is outside some disk), and there is an integer n such that exists and is a nonzero complex number.

The complex plane extended by a point at infinity is called the Riemann sphere.

If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros.

Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.

For a general discussion of zeros and poles of such functions, see Pole–zero plot § Continuous-time systems.

The simplest examples of such curves are the complex plane and the Riemann surface.

This extension is done by transferring structures and properties through charts, which are analytic isomorphisms.

meromorphic) in a neighbourhood of a point z of M if there is a chart

If the curve is compact, and the function f is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros.

This is one of the basic facts that are involved in Riemann–Roch theorem.