Why a holomorphic function is infinitely differentiable just because of satisfying the Cauchy Riemann equations, but on the other side, a two variable real function that is twice differentiable is not infinitely differentiable?
I'm asking this for two reasons:
1) $\mathbb{R}^2$ is somehow an analog to the complex plane (they are isomorphic).
2) Holomorphic means that is differentiable in any direction of the complex plane, so why if a function is differentiable in any direction of the real plane then it is not infinitely differentiable?
Best Answer
Even though $\mathbb R^2$ and $\mathbb C$ are isomorphic as real vector spaces, they are very different in some algebraic respects, which crucially influence the notion of differentiability. Recall the key idea that a function is differentiable at a point if it has a best linear approximation (more precisely, a constant plus a linear transformation) near that point.
In the context of functions $\mathbb R^2\to\mathbb R^2$, "linear transformation" means a transformation that respects addition of vectors and multiplication by real scalars. In other words, the transformation respects the real vector space structure of $\mathbb R^2$. It is well known that such linear transformations are given by $2\times2$ real matrices (once one has chosen a basis for $\mathbb R^2$).
In the context of functions $\mathbb C\to\mathbb C$, on the other hand, "linear transformation" means a transformation that respects addition of vectors and multiplication by complex scalars. In other words, the transformation respects the complex vector space structure of $\mathbb C$. It is well known that such linear transformations are just multiplication by a single complex number. That's much more restrictive than multiplying by an arbitrary $2\times 2$ real matrix.
Specifically, if we use $\{1,i\}$ as our basis for $\mathbb R^2$, then the $2\times 2$ real matrices that correspond to complex linear transformations are just those of the form $\pmatrix{a & b\\-b & a}$. Because complex linear transformations are a very special sort of real linear transformations, complex differentiable functions are a very special sort of real differentiable functions. That "specialness" ultimately accounts for all the miraculous consequences of complex differentiability.