Graduate Complex analysis -- Essential singularity

[LagrangeEuler]

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?

 

[fresh_42]

Have you google "essential singularity + pdf"?

 

[LagrangeEuler]

Yes and I did not find any other example.

 

[fresh_42]

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.

 

[LagrangeEuler]

Yes, I know that. But I do not know how to find those examples.

 

[fresh_42]

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.

 

[FactChecker]

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.