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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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Complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.
Laurent series
In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.
Residue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.