# For the function $f\left( z \right) = \sin \left( {\frac{1}{z}} \right),\;z = 0$ is a:

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For the function $f\left( z \right) = \sin \left( {\frac{1}{z}} \right),\;z = 0$ is a:
1. Removable singularity
2. Simple pole
3. Branch point
4. Essential singularity

by (19.3k points)
selected

Correct Answer - Option 4 : Essential singularity

$f\left( z \right) = \sin \left( {\frac{1}{z}} \right)$

We know that,

$\sin z = z - \frac{{{z^3}}}{{3!}} + \frac{{{z^5}}}{{5!}} \ldots$

$f\left( z \right) = \sin \left( {\frac{1}{z}} \right) = \frac{1}{z} - \frac{1}{{3!{z^3}}} + \frac{1}{{5!{z^5}}} \ldots$

If the number of negative powers of (z - a) in the f(z) is negative, then z = a is called an essential singularity.

∴ f(z) has infinite negative powers of z = 0 and z = 0 is an essential singularity.
The passage contains a question about the nature of a singularity for a given complex function, and it provides the correct answer among multiple choices.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# For the function $f\left( z \right) = \sin \left( {\frac{1}{z}} \right),\;z = 0$ is a:

86 views

closed
For the function $f\left( z \right) = \sin \left( {\frac{1}{z}} \right),\;z = 0$ is a:
1. Removable singularity
2. Simple pole
3. Branch point
4. Essential singularity

by (19.3k points)
selected

Correct Answer - Option 4 : Essential singularity

$f\left( z \right) = \sin \left( {\frac{1}{z}} \right)$

We know that,

$\sin z = z - \frac{{{z^3}}}{{3!}} + \frac{{{z^5}}}{{5!}} \ldots$

$f\left( z \right) = \sin \left( {\frac{1}{z}} \right) = \frac{1}{z} - \frac{1}{{3!{z^3}}} + \frac{1}{{5!{z^5}}} \ldots$

If the number of negative powers of (z - a) in the f(z) is negative, then z = a is called an essential singularity.

∴ f(z) has infinite negative powers of z = 0 and z = 0 is an essential singularity.
user
The passage contains a question about the nature of a singularity for a given complex function, and it provides the correct answer among multiple choices.

TYPE: SINGLE_QUESTION_WITH_ANSWER