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/zⁿ 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:

• Poles of 6/((1 – z)(5 – 9z)) »
• Poles of 6/((1 – z)(5 – 9z)) when |z| ≤ 5/9 »
• Poles of gamma[z] »

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:

• Residues of x! »
• Residues of 1/(e^z – 1) »
• Residues of tan[z] for 2 < |z| < 5 »

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!