Pulse-width modulation

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 $\scriptstyle {\frac {1}{z^{n}}}$ 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.

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 $\scriptstyle z\,=\,0$.

• The function

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

has a pole of order 2 at $\scriptstyle z\,=\,5$ and a pole of order 3 at $\scriptstyle z\,=\,-7$.

• The function

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

has poles of order 1 at $\scriptstyle z\,=\,2\pi ni{\text{ for }}n\,=\,\dots ,\,-1,\,0,\,1,\,\dots .$ To see that, write $\scriptstyle e^{z}$ 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.

See also

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

External links

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• Module for Zeros and Poles by John H. Mathews