Complex differentiable implies analytic

Question

The only proof of this that I am aware of, uses complex line integration. This tool allows for elegant proof, but has deprived me of any intuitive or geometric understanding of why this is the case. Is there a deeper, more enlightening reason for why this is true?

Answer 1

Deeper - yes, more intuitive - I do not think so. This is a special case of the general principle that solutions of elliptic PDE with analytic coefficients are analytic (=expand into convergent power series). The elliptic PDE in question is of course the Cauchy-Riemann equation, or the Laplace equation that is derived from it. For the Laplace equation this is called ``Weyl's lemma''. For general (non-linear) elliptic equations of second order, this was one of the Hilbert problems, solved by Serge Bernstein.

All these proofs are quite advanced, and I cannot say that they are more intuitive than Cauchy's proof for the Cauchy-Riemann equations. So this was indeed a great discovery of Cauchy: a fact that is very far from being intuitively clear.