Pole (complex analysis)

Definition

Formally, suppose U is an open subset of the complex plane C, p is an element of U and f : U \ {p} → C is a function which is holomorphic over its domain. If there exists a holomorphic function g : U → C, such that g(p) is nonzero, and a positive integer n, such that for all z in U \ {p}

f ( z ) = g ( z ) ( z − p ) n

holds, then p 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 p, then necessarily g(p) ≠ 0 for the function g in the above expression. So we can put

f ( z ) = 1 h ( z )

for some h that is holomorphic in an open neighborhood of p and has a zero of order n at p. 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 ) = a − n ( z − p ) n + ⋯ + a − 1 ( z − p ) + ∑ k ≥ 0 a k ( z − p ) k .

This is a Laurent series with finite principal part. The holomorphic function ∑ k ≥ 0 a k ( z − p ) k (on U) is called the regular part of f. So the point p is a pole of order n of f if and only if all the terms in the Laurent series expansion of f around p below degree −n vanish and the term in degree −n is not zero.

Pole at infinity

A complex function can be defined as having a pole at the point at infinity. In this case U has to be a neighborhood of infinity, such as the exterior of any closed ball. To use the previous definition, a meaning for g being holomorphic at ∞ is needed. Alternately, a definition can be given starting from the definition at a finite point by suitably mapping the point at infinity to a finite point. The map z ↦ 1 z does that. Then, by definition, a function f holomorphic in a neighborhood of infinity has a pole at infinity if the function f ( 1 z ) (which will be holomorphic in a neighborhood of z = 0 ), has a pole at z = 0 , the order of which will be regarded as the order of the pole of f at infinity.

Pole of a function on a complex manifold

In general, having a function f : M → C that is holomorphic in a neighborhood, U , of the point a , in the complex manifold M, it is said that f has a pole at a of order n if, having a chart ϕ : U → C , the function f ∘ ϕ − 1 : C → C has a pole of order n at ϕ ( 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 ϕ ( z ) = 1 z .

Examples

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 poles of order 1 at z = 2 π n i  for  n = … , − 1 , 0 , 1 , … . To see that, write e z in Taylor series around the origin.

The function

has a single pole at infinity of order 1.

Terminology and generalizations

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.