 Pole (complex analysis)

In the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of at z = 0. This means that, in particular, a pole of the function f(z) is a point a such that f(z) approaches infinity as z approaches a.
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Definition
Formally, suppose U is an open subset of the complex plane C, a is an element of U and f : U \ {a} → C is a function which is holomorphic over its domain. If there exists a holomorphic function g : U → C and a positive integer n, such that for all z in U \ {a}
holds, then a is called a pole of f. The smallest such n is called the order of the pole. A pole of order 1 is called a simple pole.
A few authors allow the order of a pole to be zero, in which case a pole of order zero is either a regular point or a removable singularity. However, it is more usual to require the order of a pole to be positive.
From above several equivalent characterizations can be deduced:
If n is the order of pole a, then necessarily g(a) ≠ 0 for the function g in the above expression. So we can put
for some h that is holomorphic in an open neighborhood of a and has a zero of order n at a. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions.
Also, by the holomorphy of g, f can be expressed as:
This is a Laurent series with finite principal part. The holomorphic function (on U) is called the regular part of f. So the point a is a pole of order n of f if and only if all the terms in the Laurent series expansion of f around a below degree −n vanish and the term in degree −n is not zero.
Pole at infinity
It can be defined for a complex function the notion of having a pole at the point at infinity. In this case U has to be a neighborhood of infinity. For example, the exterior of any closed ball. Now, for using the previous definition a meaning for g being holomorphic at ∞ should be given and also for the notion of "having" a zero at infinity as does at the finite point a. Instead a definition can be given starting from the definition at a finite point by "bringing" the point at infinity to a finite point. The map does that. Then, by definition, a function, f, holomorphic in a neighborhood of infinity has a pole at infinity if the function (which will be holomorphic in a neighborhood of ), has a pole at , the order of which will be taken as the order of the pole at infinity.
Pole of a function on a complex manifold
In general, having a function that is holomorphic in a neighborhood, , of the point , in the complex manifold M, it is said that f has a pole at a of order n if, having a chart , the function has a pole of order n at (which can be taken as being zero if a convenient choice of the chart is made). ] The pole at infinity is the simplest nontrivial example of this definition in which M is taken to be the Riemann sphere and the chart is taken to be .
Examples
 The function
 has a pole of order 1 or simple pole at .
 The function
 has a pole of order 2 at and a pole of order 3 at .
 The function
 has poles of order 1 at To see that, write in Taylor series around the origin.
 The function

 f(z) = z
 has a single pole at infinity of order 1.
Terminology and generalisations
If the first derivative of a function f has a simple pole at a, then a is a branch point of f. (The converse need not be true).
A nonremovable singularity that is not a pole or a branch point is called an essential singularity.
A complex function which is holomorphic except for some isolated singularities and whose only singularities are poles is called meromorphic.
See also
 Control theory#Stability
 Filter design
 Filter (signal processing)
 Nyquist stability criterion
 Pole–zero plot
 Residue (complex analysis)
 Zero (complex analysis)
External links
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