Complex analysis -- Essential singularity

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Discussion Overview

The discussion revolves around identifying examples of essential singularities in complex analysis, specifically seeking functions beyond the well-known example of f(z) = e^{1/z}. Participants also inquire about resources for finding such examples.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant requests additional examples of essential singularities and references a desire for a book containing such examples.
  • Another participant suggests searching for resources online, specifically mentioning the phrase "essential singularity + pdf."
  • One participant explains that an essential singularity occurs when an infinite number of terms with negative exponents remain in the Laurent series expansion, implying that many examples can be generated from this concept.
  • Another participant expresses uncertainty about how to find specific examples of essential singularities despite understanding the theoretical framework.
  • It is noted that functions with an infinite Taylor series can lead to an infinite negative part of the Laurent series when applying the transformation z → 1/z, with sine, cosine, and logarithm mentioned as potential examples.
  • A later reply emphasizes that the original Taylor series must have an infinite radius of convergence for the examples to be valid.

Areas of Agreement / Disagreement

Participants do not reach a consensus on specific examples of essential singularities, and there is ongoing uncertainty regarding how to identify or find these examples.

Contextual Notes

There are limitations regarding the clarity of definitions and the dependence on the properties of the functions discussed, particularly concerning the radius of convergence of Taylor series.

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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