I'm learning about different types of singularities:
1) removable singularities.
2) poles.
3) essential singularities.
I think I understand 1 and 2 but I don't really get the 3rd one, could someone please explain me?
1) For a removable singularity $|f|$ is bounded near $z_o$ so as an example we'd have a removable singularity at $z_0=0$ for $\frac{sin(z)}{z}$.
2) For a pole $|f|$ goes to infinity near $z_o$ , so as an example we'd have a pole at $z_0=0$ for $\frac{1}{z}$.
3) For an essential singularity $|f|$ doesn't go to infinity near $z_o$ nor is bounded here. In the book they give the example: we'd have an essential singularity at $z_0=0$ for $e^{\frac{1}{z}}$.
I don't get this last one, since in my head: (I know it's not mathematically correct what I'm gonna write but I do it just to explain my problem) $e^{\frac{1}{0}}=e^\infty=\infty$
Thanks in advance.
Best Answer
Note that a function $f(z)$ can be expanded as a Laurent series about a singularity $z_0$ as: $$f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n + \sum_{n=1}^{\infty} \frac{b_n}{(z-z_0)^n}$$
Hence, the given function: $$f(z) = e^{\frac1{z}} = \sum_{n=0}^{\infty} \frac{1}{n!z^n}$$ has $z=0$ as an essential singularity.