Pole (complex analysis)

The absolute value of the Gamma function. This shows that a function becomes infinite at the poles (left, for negative real values). 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 $\scriptstyle \phi(z)\, = \,\frac{1}{z}$ .

Examples

The function

f(z) = \frac{3}{z}

has a pole of order 1 or simple pole at

z= 0

The function

f(z) = \frac{z+2}{(z-5)^2(z+7)^3}

has a pole of order 2 at

z=5

and a pole of order 3 at

z = -7

The function

f(z) = \frac{z-4}{e^z-1}

has poles of order 1 at

z\, = \,2\pi ni\text{ for } n\, = \,\dots,\, -1,\, 0,\, 1,\, \dots.

To see that, write

e^z

in Taylor series around the origin.

The function

f(z) = z

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.

External links

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