Question

types of singularities in complex analysis​

Answer 1

Answer:

In complex analysis, there are several classes of singularities .These include the isolated singularities ,the nonisolated singularties and the branch points

Answer 2

Answer:

Some complex functions are analytic at all points of a bounded domain except at a finite number of points. These points are called singular points or singularities.

The point $z=\alpha$ is called a singular point or singularities of the complex function f(z) if f(z) is not analytic at point $\alpha$ but every neighborhood of $\alpha$ is analytic.

e.g., the function $f(z)=\frac{1}{2-z}$ is analytic for all value of z except $z=2$. Thus $z=2$ is a singular point of $f(z)$.

Various types of singularities in complex analysis are discussed below:

Isolated Singularities:

A function $f(z)$ is said to have an isolated singularity at $z=\alpha$ if it is analytic in the deleted neighborhood of $\alpha$ i.e., $0 < |z-\alpha | < \delta$.

Thus if $\alpha$ is an isolated singularities of $f(z)$ then there is no other singularities in the neighborhood of $\alpha$.

Isolated singularities are of 3 types.

(i) Removable Singularities:

If the principal part of the Laurent series of a function $f(z)$ at $z=\alpha$ has no term, then $z=\alpha$ is called  a removable singularity.

e.g., $z=0$ is a removable singularity of $f(z)=\frac{sin z}{z}$

(ii) Pole:

If the  principal part of the Laurent series of a function $f(z)$ containing finite number of terms, then the singularity $z=\alpha$ is called a pole.

e.g., $z=0$ is a pole of order 3 of $f(z)=\frac{e^z}{z^3}$

A pole of order 1 is called as single pole.

(iii) Essential Singularities:

If a function $f(z)$ is expressed as Laurent series about $z=\alpha$ and its principal part contains infinite number of terms then the singularity $z=\alpha$ is called an essential singularity.

e.g., $z=0$ is an essential singularity of $f(z)=z^2.\frac{1}{e^z}$