Complex analysis -- Essential singularity

  • #1
LagrangeEuler
717
20
Can you give me two more examples for essential singularity except [tex]f(z)=e^{\frac{1}{z}}[/tex]? And also a book where I can find those examples?
 
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  • #2
Have you google "essential singularity + pdf"?
 
  • #3
Yes and I did not find any other example.
 
  • #4
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.
 
  • #5
Yes, I know that. But I do not know how to find those examples.
 
  • #6
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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  • #7
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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