August 13, 2012 Leave a comment
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 which is is undefined at some point . We can expand it as a Laurent series, and get something like:
It may be that we can make arbitrarily negative and still have 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
In a better situation, it might be that is nonzero, but with any smaller (more negative) index, is zero. Then, we would say that has a pole at of order . 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!