How to find Removable Singularity?

A removable singularity is a point at which a function is not defined, but can be defined in such a way that the function becomes analytic (i.e., smooth and continuous) at that point. Here's how you can determine if a singularity is removable:

Steps to find a removable singularity:
Identify the singularity:
Start by identifying the point where the function is undefined or has a singularity. This is typically where the denominator of a rational function or the argument of a logarithmic or trigonometric function causes a problem.

Check if the limit exists:
To determine whether the singularity is removable, you need to check if the limit of the function exists as it approaches the singularity. If the function approaches a finite value as it nears the point, the singularity is removable. Specifically, if the limit:

lim (z → z₀) f(z)

exists and is finite, then the singularity at z₀ is removable.

Check for a factorable form:
If the singularity is a pole or essential singularity, the limit will not exist in a finite form. However, for a removable singularity, the function often has a factorable form, where the problematic term in the denominator can be canceled out by the numerator. For example, if you have a function like:

f(z) = (z - z₀)g(z)

and g(z) is analytic at z₀, then the singularity at z₀ is removable.

Define the function at the singularity:
If the limit exists and is finite, define the function at the singularity to make it continuous. This definition should match the limit you calculated. This will ensure that the function is analytic at the singularity.

Example:
Consider the function:

f(z) = (sin z) / z

At z = 0, this function has a singularity. To determine if it's removable, we compute the limit:

lim (z → 0) (sin z) / z

We know that:

lim (z → 0) (sin z) / z = 1

Since the limit exists and is finite, the singularity at z = 0 is removable. If we redefine the function at z = 0 as:

f(0) = 1

then f(z) becomes analytic at z = 0, removing the singularity.

Conclusion:
A singularity is removable if the limit of the function exists and is finite as the point is approached. You can redefine the function at that point to make it continuous and analytic.