Singularities, essential singularities, poles, simple poles

Question

Could someone possible explain the differences between each of these;

Singularities, essential singularities, poles, simple poles.

I understand the concept and how to use them in order to work out the residue at each point, however, done fully understand what the difference is for each of these

As far as i understand a simple pole is a singularity of order $1$?

then we have poles of order $n$ which aren't simple?

not too sure about essential singularity

Answer 1

There are three kinds of singularities.

Removable singularity, which can be extended to a holomorphic function over that point.

Poles, which is removable after multiplying some $(z-a)^n$. The smallest $n$ is called the order of the pole, when $n=1$, it is called simple.

Essential singularity: neither of the above. For example $g(z)=e^{1/z}$ since $|g(z)z^l|$ is never bounded near $0$.

Answer 2

Singularity:

$\quad$ A point $a$ is said to be a singular point of a function $f$ if

i) f is not analytic at $a$ and

ii) if we can find a neighborhood of $f(a)$ such that there exists a point $b$ in which $f$ is analytic.

Essential Singularity:

$\quad$ A point $a$ is said to be a essential singular point of a function $f$ if

i) f is not analytic at $a$ and

ii) if every neighborhood of $f(a)$ contains infinte number of points in which $f$ is analytic.

Poles:

a point $a$ is said to be a pole if

i)it is a essential singularity and

ii)$\lim_{z \to a} f(z) = \infty$

A pole of order 1 is simple pole and double pole if it is order 2.