Do Residues and Singularities Define Functions Like tanh(z) and tan(z)?

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SUMMARY

Residues exist for holomorphic functions, and both tanh(z) and tan(z) exhibit essential isolated singularities due to their infinite series expansions. The residue for both functions at their respective singularities is calculated to be 0. This conclusion is drawn from the definitions of residues and holomorphic functions, emphasizing that residues are specific to points rather than functions as a whole.

PREREQUISITES
  • Understanding of complex analysis concepts, particularly residues and singularities.
  • Knowledge of holomorphic functions and their properties.
  • Familiarity with the definitions and calculations involving essential isolated singularities.
  • Experience with series expansions in complex functions.
NEXT STEPS
  • Study the properties of holomorphic functions in detail.
  • Learn about the classification of singularities in complex analysis.
  • Explore the calculation of residues at isolated singularities using Laurent series.
  • Investigate the behavior of tanh(z) and tan(z) near their singular points.
USEFUL FOR

Students and professionals in mathematics, particularly those focusing on complex analysis, as well as educators teaching the concepts of residues and singularities in advanced calculus courses.

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Homework Statement


Do residues exist only for holomorphic function?

Classify the singularity and calculate the residue of tanh(z) and tan(z)

The Attempt at a Solution


For both
Essential isolated singularity because the numerator has an infinite number of terms.

Residue = 0.
 
Last edited:
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pivoxa15 said:

Homework Statement


Do residues exist only for holomorphic function?

Classify the singularity and calculate the residue of tanh(z) and tan(z)




The Attempt at a Solution


For both
Essential isolated singularity because the numerator has an infinite number of terms.

Residue = 0.
What are the definitions of "residue" and "holomorphic function"?

At what point do tanh(z) and tan(z) have an isolated singularity? It doesn't make sense to talk about the "residue" of a function. A function may have a "residue" at a specific point.
 

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