#differentiability

The first change we encounter as we move to complex functions is the geometric interpretation of a derivative. My postdoc has teased and teased about this point. In real analysis, the derivative is just the slope of a tangent line/plane/etc (depending on the dimension of the space). In complex analysis, we get two problems. First of all, a complex function maps complex numbers to complex numbers, so it maps a two-dimensional object to a different two-dimensional object. As such, we'd need four dimensions to graph a complex function the same way we graph real functions. Obviously, our universe is limited to three dimensions, so we can't properly visualise a complex function. Instead, we can think about lines/regions in the domain and their image under the function. While this kind of method helps in seeing how the function warps the complex plane, it doesn't help us with our geometric interpretation of the derivative. Our second problem is that the derivative of a function at a particular point is just the limit of the ratio of the change in the function over the change in the parametre. However, the function and parametre are both complex-valued, so the limit (ie derivative) will be a complex number. What does it mean to get a complex number as a derivative anyways? Well...the postdoc has mentioned this problem quite a few times during the past few weeks, and he keeps teasing that we'll get a "geometric" interpretation further into the class. Okay, postdoc. We're waiting. It's your move.

Here's what we've got so far: the derivative of a function exists at a point if the first principles limit exists at the point. Easy enough. Here's a cool result that we get if the derivative exists: let's parametrise f(z) = f(x + iy) = u(x, y) + iv(x, y) where u and v are real-valued functions. Let's say that the derivative exists at c = a + ib. Then, we get the following:

the partial derivatives of u, v, with respect to x and y will exist at c = a + ib.

at c = a + ib, the following hold: ux = vy and uy = -vx.

then, we can write f' = ux + ivx = vy - iuy.

These equalities are called the cauchy-riemann equations, and voila! We see our cauchybro popping up again. In fact, according to wikipedia, Cauchy was one cool dude because he has had more theorems/concepts named after him than any other dude. The proof of the cauchy-riemann equations follow from the fact that if the limit f' exists at c, then it exists no matter how we approach c; so we can approach it "real-ly" by letting y = 0, or we can approach it "imaginary-ly" by letting x = 0. This way, we get the third bullet and thus the equalities listed in the second bullet.

Now what about the converse? ie, can we use these ideas to find a sufficient condition for differentiability? Turns out, yes. Yes, we can. Let's suppose we want f to be differentiable at c = a + ib. Then, we need:

all the partials of u, v, with respect to x and y exist and are continuous in some neighbourhood of c.

at the point c, the cauchy-riemann equations hold.

The proof of this way around sucked hard. If you ever need it, just read the book. ugh.

So another thing that happens in complex analysis is that we can get functions that are not differentiable anywhere EXCEPT for one point. That's quite nasty and annoying and, frankly, it's not even worth it to be differentiable at only one point. So a nice property for a function to have is that it is analytic. A function is analytic at c = a + ib if the function is everywhere differentiable in some neighbourhood of c. We see that analytic functions are in fact infinitely differentiable, which is a beautiful property to have, isn't it?

How do we guarantee a function is analytic on some open domain D? Hey, I see cauchy cropping up again. A function is analytic iff:

u and v are harmonic functions on D.

u and v satisfy the cauchy-riemann equations on D.

What does it mean for u and v to be harmonic functions? The first and second order derivatives need to exist and be continuous, and the sum of the second order partials need to equal 0. In this case, we need uxx + uyy = 0 for u to be harmonic. If we get that u and v are harmonic and they satisfy the cauchy-riemann equations, then we call v the harmonic conjugate of u. If v is the harmonic conjugate of u, we can observe that -u is the harmonic conjugate of v.

If a function is analytic across its whole domain, then we call it an entire function.