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

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