Why are differentiable complex functions infinitely differentiable?

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:RR 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:CC 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

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