Pole

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 z = a such that f(z) approaches infinity uniformly as z approaches a.

The absolute value of the Gamma function. This shows that a function becomes infinite at the poles (left). On the right, the Gamma function does not have poles, it just increases quickly.

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 ∑_k≥0a_k (z - a)^k (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 vanishes and the term in degree −n is not zero.

Examples

• The function

has no poles (the denominator never becomes 0).

• The function

has a pole of order 1 or simple pole at z = 0.

• The function

has a pole of order 2 at z = 5 and a pole of order 3 at z = − 7.

• The function

has a pole of order 1 at z = 0. To see that, it is necessary to write e^z in Taylor series around the origin.

Remarks

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 non-removable 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

• Zero (complex analysis)
• Residue (complex analysis)
• Filter (signal processing)
• Control theory#Stability
• Pole–zero plot
• Filter design
• Nyquist stability criterion

External links

• Weisstein, Eric W., "Pole" from MathWorld.
• Module for Zeros and Poles by John H. Mathews