Pole (complex analysis)

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, 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 $\scriptstyle \sum _{k\,\geq \,0}a_{k}(z\,-\,a)^{k}$ (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

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.