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Problem 5

# Discuss the singularities of $$f(z)=\frac{z^{3}\left(z^{2}-1\right)(z-2)^{2}}{\sin ^{2}(\pi z)} e^{1 / z^{2}}$$ Classify which of these are poles, removable singularities and essential singularity.

Expert verified
All the singularities of the function are essential singularities at points $$z = n$$, where $$n$$ is an integer.
See the step by step solution

## Step 1: Determine where the function is undefined

The function is undefined when the denominator, $$\sin^{2}(\pi z)$$, is zero. Thus, the solution is $$z = n$$ where $$n$$ is an integer.

## Step 2: Classify the singularities

In order to classify the singularities, we apply the following rules: If the function's limit as $$z$$ approaches the singular point exists, it's a removable singularity. If the limit is infinity, it's a pole. If neither of these, it's an essential singularity. For $$z = n$$ where $$n$$ is an integer, the limit does not exist due to the $$e^{1/z^{2}}$$ term in the function, thus all of the singularities are essential singularities.

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