When I studied complex analysis, I could never understand how once-differentiable complex functions could be possibly be infinitely differentiable. After all, this doesn’t hold for functions from R2 to R2. Can anyone explain what is different about complex numbers?

**Answer**

As Akhil mentions, the keyword is elliptic regularity. Since I don’t know anything about this, let me just say some low-level things and maybe they’ll make sense to you.

A differentiable function f:R→R can be thought of as a function which behaves locally like a linear function f(x)=ax+b. So, very roughly, it is a collection of tiny vectors which fit together. These tiny vectors can, however, fit together in a very erratic manner. That’s because since you only have to fit one vector to the two vectors that are its neighbors, there is a lot of room for bad behavior.

A differentiable function f:C→C has to satisfy a much more stringent requirement: locally, it has to behave like a linear function f(z)=az+b where z,a,b are complex, which is a **rotation** (and scale, and translation). So, very roughly, it is a collection of tiny rotations which fit together. Now one rotation has a continuum of neighbors to worry about, and it becomes much harder for erratic behavior to persist.

**Attribution***Source : Link , Question Author : Casebash , Answer Author : Qiaochu Yuan*