Singularity

In complex analysis, a singularity (or singular point) of a complex function is a point $z=z_0$ where it fails to be analytic. We’re interested in classifying isolated singular points depending on the principal part of a function’s Laurent series:

• If the principal part is zero, i.e., all the negative $k$-th coefficients are zero, then the point is a removable singularity.
• If it contains a finite number of nonzero parts, then the point is called a pole. If it’s a pole of order 1, then it’s called a simple pole.¹ For an $n$-th negative coefficient, we have a pole of order n.
• If the principal part has infinitely many non-zero terms, then the point is an essential singularity.

Isolated singularity

We say that $f(z)$ has an isolated singular point at $z_0$ if $f(z)$ is analytic for $0<|z-z_0|<R$ for some $R>0$.

For example:

$$f(z)=\frac{1}{(z-1)(z-3)}$$

has two isolated singularities at $z=1,3$. However,

$$\text{Ln}(z)={\log }_e|z|+i\text{Arg}(z)$$

is not continuous for $\text{Re}(z)\le 0$. $\text{Arg}(z)$ is not continuous at every $z_0$ on the negative real axis, so $f(z)$ is not analytic in $(-\infty ,0]$, i.e., these aren’t isolated singularities.

Footnotes

1. Prof Nachman has a joke he likes to tell. The gist of it is that on a plane, the pilots fall asleep. The flight attendants ask the passengers if anyone is a pilot and able to fly a plane. Coincidentally, a Polish man says he flew planes in WWI — so he volunteers to go into the cockpit and have a look. Unfortunately planes have progressed quite a bit since WWI, so he’s rightfully overwhelmed and exclaims “I’m just a simple pole in a complex plane”. Haha. Hilarious. ↩