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!

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