# New Features for Students of Complex Analysis

Why did the mathematician name his dog Cauchy? Because he left a residue at every pole!

But could the mathematician find the poles and their residues for a given function? He certainly could, with the help of Wolfram|Alpha.

We are proud to announce that Wolfram|Alpha has added residues and poles to its ever-expanding library of mathematical data that it can compute! To showcase this behavior, let’s first recall just what a pole is.

In the study of complex analysis, a pole is a singularity of a function where the function behaves like 1/*z*^{n} at *z* == 0 .

More technically, a point *z*_0 is a pole of a function if the Laurent series expansion of the function about the point *z*_0 has only finitely many terms with a negative degree of *z* – *z*_0. As an example, let’s look at the Laurent expansion of 1/Sin[*z*] at *z* == 2 Pi:

We can see here that there is only one “*z* – 2 Pi” term with a negative degree, and that is the term 1/(*z* – 2 Pi). This lets us know that 1/Sin[*z*] has a pole at *z* == 2 Pi.

Where else does this function have poles? Let’s ask Wolfram|Alpha!

In fact, let’s ask Wolfram|Alpha about a few other functions:

In complex analysis, we are particularly interested in the residue of a function at its poles. Looking at the Laurent expansion of a function about a pole, *z*_0, we can easily see its residue at *z*_0, because the residue is the coefficient of the (*z* – *z*_0)^{-1} term.

In our example above, where we looked at the Laurent expansion of 1/Sin[*z*], we found that it had exactly one term with a negative exponent, and that term was 1/(*z* – 2 Pi). As we look at this term now, we see that the residue for 1/Sin[*z*] is 1. Let’s double check our work using Wolfram|Alpha:

More generally, what are the residues of 1/Sin[*z*]?

And, for fun, let’s look at a few others:

We are proud to announce that Wolfram|Alpha is now a growing resource for students of complex analysis. We hope that you enjoy exploring this new addition to Wolfram|Alpha!