Complex Differentiability Part 2: What else do we need?

September 14, 2010

This is going to be a quick post, and a wrap-up to the other post I made about this.  I realized that, besides the Cauchy-Riemann equations being satisfied, very little else needs to happen such that a function is complex differentiable at a point $z_{0}$.  What else, specifically?  Let’s make this a nice little theorem.

Theorem:  A function $f(z)$ is complex differentiable at $z_{0}$ if it satisfies the Cauchy-Riemann equations, if the CR-equations exist at the point $z_{0}$, and if the CR-equations are continuous in a neighborhood of $z_{0}$.

I’m not going to do the proof, since it’s one of those "analysis" type proofs that I don’t think makes anything clear to anyone.  Essentially, the point of the theorem is that, not only do the partials have to exist and satisfy CR, but they also have to be relatively nice in order for the derivative to exist (if a function is not even continuous, what hope do we have of it being differentiable?); this is that "continuous" part.  Since continuity is a local property, it must be defined on a neighborhood — hence the neighborhood part of the theorem.  Essentially, then, we just need to check a few things.  Given some $f$, we

• Check that the partials exist at the point $z_{0}$ and that they’re continuous at the point $z_{0}$.  This is "usually" relatively easy, since partials are usually either polynomials or rational functions in practice.
• Check that the partials satisfy the CR-equations.  This is also relatively easy, since we’ve just computed all of them.

And that’s really it.  In fact, you can really do that in any order, but if you do’em all, and they all check out, your function is continuous at $z_{0}$.  Good for you!

Note that the other way around makes a few things trivial.  If we have that a function is complex differentiable at a point, we automatically know that it is continuous at that point and that its partials are also continuous at that point.  This also will satisfy the CR-equations by what we proved last time.

Okay?  Okay.  Good.  Let’s last note (this is really what you should take away from all this) that it’s not enough to say that a function satisfies the CR-equations for it to be differentiable at a point; we need existence and continuity of the partials at that point as well.