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 \( \scriptstyle \frac{1}{z^n} \) at z = 0. For a pole of the function f(z) at point a the function approaches infinity as z approaches a.

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}

\( f(z) = \frac{g(z)}{(z-a)^n} \)

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

\( f(z) = \frac{1}{h(z)} \)

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:

\( f(z) = \frac{a_{-n}}{ (z - a)^n } + \cdots + \frac{a_{-1}}{ (z - a) } + \sum_{k\, \geq \,0} a_k (z - a)^k. \)

This is a Laurent series with finite principal part. The holomorphic function \( \scriptstyle \sum_{k\,\ge\,0} a_k(z\, - \,a)^k \) (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 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 \( \scriptstyle z\, - \,a \) 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 \( \scriptstyle z\, \mapsto \,\frac{1}{z} \) does that. Then, by definition, a function, f, holomorphic in a neighborhood of infinity has a pole at infinity if the function \( \scriptstyle f(\frac{1}{z}) \) (which will be holomorphic in a neighborhood of \( \scriptstyle z\,=\,0) \) , has a pole at \( \scriptstyle z\,=\,0 \), 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 \( \scriptstyle f:\; M\, \rightarrow \,\mathbb{C} \) that is holomorphic in a neighborhood, \scriptstyle U, of the point \(\scriptstyle a \), in the complex manifold M, it is said that f has a pole at a of order n if, having a chart \(\scriptstyle \phi:\; U\, \rightarrow \,\mathbb{C} \), the function \( \scriptstyle f\, \circ \,\phi^{-1}:\; \mathbb{C}\, \rightarrow \,\mathbb{C} \) has a pole of order n at \scriptstyle \phi(a) (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 \( \scriptstyle \phi(z)\, = \,\frac{1}{z} \).

Examples

The function

\( f(z) = \frac{3}{z} \)

has a pole of order 1 or simple pole at \( \scriptstyle z\, = \,0 \).

The function

\( f(z) = \frac{z+2}{(z-5)^2(z+7)^3} \)

has a pole of order 2 at \( \scriptstyle z\, = \,5 \) and a pole of order 3 at \scriptstyle z\, = \,-7.

The function

\( f(z) = \frac{z-4}{e^z-1} \)

has poles of order 1 at \( \scriptstyle z\, = \,2\pi ni \text{ for } n\, = \,\dots,\, -1,\, 0,\, 1,\, \dots. \) To see that, write \( \scriptstyle e^z \) 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 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

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