Complex analysis is filled with theorems that seem too good to be true. One is that if a complex function is once differentiable, it’s infinitely differentiable. How can that be? Someone asked this on math.stackexchange and this was my answer.
The existence of a complex derivative means that locally a function can only rotate and expand. That is, in the limit, disks are mapped to disks. This rigidity is what makes a complex differentiable function infinitely differentiable, and even more, analytic.
For a complex derivative to exist, it has to exist and have the same value for all ways the “h” term can go to zero in (f(z+h) – f(z))/h. In particular, h could approach 0 along any radial path, and that’s why an analytic function must map disks to disks in the limits.
By contrast, an infinitely differentiable function of two real variables could map a disk to an ellipse, stretching more in one direction than another. An analytic function can’t do that.
A smooth function of two variables could also flip a disk over, such as f(x, y) = (x, -y). An analytic function can’t do that either. That’s why complex conjugation is not an analytic function.
You might think that if complex differentiability is so restrictive, there must not be many complex differentiable functions. And yet nearly every function you’ve run across in school — trig functions, logs, polynomials, classical probability distributions, etc. — are all differentiable when extended to functions of a complex variable. According to this quote, “95 percent, if not 100 percent, of the functions studied by physics, engineering, and even mathematics students” are hypergeometric, a very special case of complex differentiable functions.