text: # 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.
explanation: The document contains a math question and its answer. The question asks about the nature of the singularity at \( z = 0 \) for the function \( f(z) = \sin \left( \frac{1}{z} \right) \). The answer provided is that \( z = 0 \) is an essential singularity.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its answer. The question asks about the nature of the singularity at \( z = 0 \) for the function \( f(z) = \sin \left( \frac{1}{z} \right) \). The answer provided is that \( z = 0 \) is an essential singularity.
  
  Label: question-and-answer