The absolute value of the Gamma function. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial 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 complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity 1/zn at z = 0. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be This means that, in particular, a pole of the function f(z) is a point z = a such that f(z) approaches infinity uniformly 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. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane If there exists a holomorphic function g : UC and a nonnegative integer n such that

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

for all z in U − {a}, then a is called a pole of f. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The smallest number n satisfying above condition is called the order of the pole. A pole of order 1 is called a simple pole. A pole of order 0 is a removable singularity. In Complex analysis, a removable singularity of a Holomorphic function is a point at which the function is ostensibly undefined but upon closer examination the

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. In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms The holomorphic function ∑k≥0ak (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 vanishes and the term in degree −n is not zero.

## Remarks

If the first derivative of a function f has a simple pole at a, then a is a branch point of f. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued (The converse need not be true).

A non-removable singularity that is not a pole or a branch point is called an essential singularity. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued In Complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior

A complex function which is holomorphic except for some isolated singularities and whose only singularities are poles is called meromorphic. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic