Question

How to find removable singularity?

Answer 1

Step 1: Explaining how to find removable singularity:

A removable singularity of a function is a point $z_0$ where the function $\mathrm{f}\left({\mathrm{z}}_0\right)$ appears to be undefined. We can remove this singularity by redefining the function, then we can say we remove the singularity.

Step 2: Writing example:

$\mathrm{f}\left(\mathrm{z}\right)=\frac{{\mathrm{z}}^2-1}{\mathrm{z}-1}$ at $\mathrm{z}=1$the function is undefined.

$\mathrm{f}\left(\mathrm{z}\right)$ has a removable singularity.

$$\begin{gathered} \mathrm{f}\left(\mathrm{z}\right)=\frac{(\mathrm{z}-1)(\mathrm{z}+1)}{(\mathrm{z}-1)}\mathbf{}\left[\because ({\mathrm{a}}^2-{\mathrm{b}}^2)=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})\right] \\ \mathrm{f}\left(\mathrm{z}\right)=(\mathrm{z}+1) \end{gathered}$$

Now the function has no singularity as we removed it.

Hence, in this way, we can find removable singularity.