Singularities in Complex Numbers

Hello reader! In previous post, we discussed about intro & basics of complex numbers. This post talks about the Singularity in the Complex numbers and its types. So, in this post, we’ll discuss about Poles, Singularity (isolated singularity, removable singularity, etc..). Let us start with what is Singularity,Later we move to its types, i.e. Singularity.For this, first we consider a function : f(z) = x+iy , which consist of real & imaginary part.

 

Singularity (Singular point):

Consider a complex function f(z) consisting of a real & imaginary part, such as, f(x) = a+bi. Then we define- Singularity is a point at which f(z) fails to be analytic, (i.e. the derivative of function f(z) does not exist.) which also has another name “Singular point”.

There are few types of singularities, we’ll study them accordingly.Isolated Singularity comes first-

Isolated Singularity:

We define isolated singularity at point z=zas, For some distance, (say) δ>0, (which is, δ=|z-z0|) such that the circle with center z0 and radius δ encloses no point other than z0.

Poles:

Pole is nothing but an isolated singularity, but with a property such that, there should be a positive integer “n” for the limit

limz-z0 (z-z0)n f(z) ≠ 0

then,  z=z0 is the pole of order n. Also, if n=1, then it is a simple pole.

There are two more types of isolated singularity they are-

Removable singularity:

It is a type of isolated singularity at z0 which can be removed by defining

f(z0) = limz-z0 f(z)

also,it can be shown that f(z) not only continuous at z0 but also analytic at z0.

Essential singularity:

An essential singularity is a singularity, which is isolated but neither pole nor removable singularity.


We have discussed about the properties of complex numbers and some of its aspects. Get a quick review to the Intro & basics of the complex numbers.