Examples of holomorphic functions
July 6, 2012 1 Comment
Since we’re going to be studying holomorphic functions, it’s probably worth seeing a few examples. I’m going to give a bunch of functions, and then prove that they’re holomorphic. I suppose the easiest would be the constant functions. Pick any , and define by . It’s clear that is continuous. It’s also a super easy computation to see that for every , so exists everywhere, meaning is holomorphic.
Well, this computation was easy, but as a general rule, explicitly computing the derivative, and checking for continuity is tedious. Instead, I’m going to make use of the theorem we proved last time:
Theorem 1 Let where is an open subset of . Write . Then if and only if all of the partial derivatives , , , and exist, and satisfy
For the constant function, we can also see that all of the partials exist, and that they satisfy the Cauchy-Riemann equations. Indeed, if , by and . Then and .
Let’s move on to something slightly more interesting. All polynomials are also holomorphic functions. The derivative is exactly what you would expect, and it’s easy to see that the partial derivatives all exist. We should check the Cauchy Riemann equations too, but to save space, I’ll leave that as an exercise.
Here’s another function which is holomorphic, but not a polynomial: Let’s try the function . We know what this means when is a real number, but more generally, what should it mean. If , I define . This jives with our normal definition of for , because the two definitions coincide when the imaginary part of is zero. Why is this holomorphic? Well, we’ve multiplied and added a bunch of functions which we know have well-defined partials, so the result has well-defined partials as well. We can just check that satisfies the Cauchy-Riemann equations. This is pretty straightforward. First I break up into it’s real and imaginary parts: where and (remember that I’m using and interchangeably). Then
which clearly satisfies the Cauchy-Riemann equations.
What about ? Is that holomorphic? Well, we need to be precise. Over all of , certainly not. It’s not even defined at zero. But remember that for holomorphic functions, I get to specify the domain. So on the domain , it turns out that is holomorphic. Again, taking , The function , can be written as
which has all partials defined except when (which is outside our domain). As usual, we write and , and check:
No list of examples would be complete without a juxtaposed non-example, so consider the conjugation function. That is, let . This function is not holomorphic on any open set. Yes, it’s continuous. Yes, it’s partial derivatives exist in every direction. But alas, it does not satisfy the Cauchy-Riemann equations, so it’s complex derivatives do not exist anywhere. We can write , where and . Taking partial derivatives, we can see that , but and . I can’t make be equal to no matter what values I plug in, so is not holomorphic anywhere.