Pole (complex analysis)

File:Gamma abs 3D.png

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.

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

This is a Laurent series with finite principal part. The holomorphic function $\scriptstyle \sum_{k\,\ge\,0} a_k(z\, - \,p)^k$ (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 $\scriptstyle z \mapsto \frac{1}{z}$ 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 regarded as the order of the pole of f 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 $\scriptstyle \phi(z)\, = \,\frac{1}{z}$ .