Chapter 9: Chapter 9

Problem 5

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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.

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All the singularities of the function are essential singularities at points $z = n$, where $n$ is an integer.

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TABLE OF CONTENTS

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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Chapter 9: Chapter 9

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.

Short Answer

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Step 1: Determine where the function is undefined

Step 2: Classify the singularities

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More chapters from the book ‘Complex Variables With Applications’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 10

Chapter 11

Chapter 12

Chapter 13

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