# 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 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*}

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 (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 does that. Then, by definition, a function *f* holomorphic in a neighborhood of infinity has a pole at infinity if the function (which will be holomorphic in a neighborhood of ), has a pole at , 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 that is holomorphic in a neighborhood, , of the point , in the complex manifold *M*, it is said that *f* has a pole at *a* of order *n* if, having a chart , the function has a pole of order *n* at (which can be taken as being zero if a convenient choice of the chart is made).
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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 .

## Examples

- The function

- The function

- The function

- has poles of order 1 at To see that, write 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.

## See also

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