# A joke

Today, a definition, then a joke.

I defined poles of meromorphic functions, but we can be a bit more descriptive. Suppose we have a meromorphic function $f$ which is is undefined at some point $z_0$. We can expand it as a Laurent series, and get something like:

$\displaystyle\sum_{-\infty}^\infty a_n(z-z_0)^n$

It may be that we can make $n$ arbitrarily negative and still have $a_n$ be nonzero. This is basically the worst situation possible. We don’t in fact call it a pole. We say that it is an essential singularity at $z_0$

In a better situation, it might be that $a_{-17}$ is nonzero, but with any smaller (more negative) index, $a_n$ is zero. Then, we would say that $f$ has a pole  at $z_0$ of order $17$. Of course there’s nothing special about seventeen.

If a pole has order 1, we say that it is a simple pole.

Now for a joke.

An airplane is on its way out of Warsaw, and the pilot suffers a heart attack and dies. A passenger is asked to navigate the plane to safety. He looks worried, so the stewardess asks “what’s wrong?” He responds “I’m just a simple Pole in a complex plane!”

Laugh, damn it!