Singular points. Isolated, removable, essential singularities. Poles.

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Def. Singular point (of an analytic function). A point at which an analytic function f(z) is not analytic, i.e. at which f '(z) fails to exist, is called a singular point or singularity of the function.

There are different types of singular points:

Isolated and non-isolated singular points. A singular point z₀ is called an isolated singular point of an analytic function f(z) if there exists a deleted ε-spherical neighborhood of z₀ that contains no singularity. If no such neighborhood can be found, z₀ is called a non-isolated singular point. Thus an isolated singular point is a singular point that stands completely by itself, embedded in regular points. See Fig. 1a where z₁, z₂ and z₃ are isolated singular points. Most singular points are isolated singular points. A non-isolated singular point is a singular point such that every deleted ε-spherical neighborhood of it contains singular points. See Fig. 1b where z₀ is the limit point of a set of singular points. Isolated singular points include poles, removable singularities, essential singularities and branch points.

Types of isolated singular points

1. Pole. An isolated singular point z₀ such that f(z) can be represented by an expression that is of the form

where n is a positive integer, f(z) is analytic at z₀, and f(z₀) ≠ 0. The integer n is called the order of the pole. If n = 1, z₀ is called a simple pole.

Example. The function

has a pole of order 3 at z = 2 and simple poles at z = -3 and z = 2.

Shown in Fig. 2 is a modulus surface of the function f(z) = 1/(z-a) defined on a region R. One sees the “pole” arising above point a in the complex plane. Thus the reason for the term “pole”. A modulus surface is obtained by affixing a Z axis to the z plane and plotting Z = |f(z)| [i.e. plotting the modulus of f(z)].

2. Removable singular point. An isolated singular point z₀ such that f can be defined, or redefined, at z₀ in such a way as to be analytic at z₀. A singular point z₀ is removable if exists.

Example. The singular point z = 0 is a removable singularity of f(z) = (sin z)/z since

3. Essential singular point. A singular point that is not a pole or removable singularity is called an essential singular point.

Example. f(z) = e ^1/(z-3) has an essential singularity at z = 3.

Singular points at infinity. The type of singularity of f(z) at z = ∞ is the same as that of f(1/w) at w = 0. Consult the following example.

Example. The function f(z) = z² has a pole of order 2 at z = ∞, since f(1/w) has a pole of order 2 at w = 0.

Using the transformation w = 1/z the point z = 0 (i.e. the origin) is mapped into w = ∞, called the point at infinity in the w plane. Similarly, we call z = ∞ the point at infinity in the z plane. To consider the behavior of f(z) at z = ∞, we let z = 1/w and examine the behavior of f(1/w) at w = 0.

References

Mathematics, Its Content, Methods and Meaning

James and James. Mathematics Dictionary

Spiegel. Complex Variables (Schaum)

Hauser. Complex Variables with Physical Applications