Graduate Complex analysis -- Essential singularity

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SUMMARY

This discussion focuses on essential singularities in complex analysis, specifically seeking examples beyond the function f(z)=e^{\frac{1}{z}}. It is established that an essential singularity occurs when an infinite number of terms in the Laurent series expansion do not vanish. Functions such as sine, cosine, and logarithm are highlighted as they can exhibit essential singularities when transformed appropriately. The requirement for the original Taylor series to have an infinite radius of convergence is also emphasized.

PREREQUISITES
  • Understanding of complex analysis concepts, particularly singularities
  • Familiarity with Taylor and Laurent series expansions
  • Knowledge of the Great Picard theorem and Casorati-Weierstraß theorem
  • Ability to manipulate complex functions and transformations
NEXT STEPS
  • Research the Great Picard theorem and its implications for essential singularities
  • Explore the Casorati-Weierstraß theorem for further understanding of singularities
  • Study the properties of infinite Taylor series and their convergence
  • Investigate additional examples of essential singularities in complex functions
USEFUL FOR

Students and professionals in mathematics, particularly those specializing in complex analysis, as well as educators seeking to deepen their understanding of essential singularities and their applications.

LagrangeEuler
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Can you give me two more examples for essential singularity except f(z)=e^{\frac{1}{z}}? And also a book where I can find those examples?
 
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Have you google "essential singularity + pdf"?
 
Yes and I did not find any other example.
 
An essential singularity exists precisely when an infinite number of terms in
$$
f(z)=\sum_{n=-\infty }^{\infty }a_n(z-z_0)^n
$$
with negative exponents do not disappear. This gives you as many examples as you wish. However, the Great Picard and Casorati-Weierstraß are pretty restrictive.
 
Yes, I know that. But I do not know how to find those examples.
 
LagrangeEuler said:
Yes, I know that. But I do not know how to find those examples.
Every function with an infinite Taylor series results in a series with an infinite negative part of the Laurent series by substituting ##z\mapsto 1/z.## Sine, cosine, logarithm, etc.
 
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fresh_42 said:
Every function with an infinite Taylor series results in a series with an infinite negative part of the Laurent series by substituting ##z\mapsto 1/z.## Sine, cosine, logarithm, etc.
Excellent. We have to add that the original Taylor series must have an infinite radius of convergence as your examples do.
 

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