[nimish]

Need to be careful about confusing smooth (infinitely differentiable) and analytic (= to Taylor series) functions.

> Any function with this property can be uniquely represented as a Taylor series expansion “centered” at a

It's fairly easy to construct tame looking functions where this isn't true.

[fractionalhare]

Nice spot, thank you! I have corrected that. I also included your comment in an acknowledgments and errata section of the doc.

[bollu]

Blessedly, in the complex analytic setting, these coincide, to they not? [Attempting to sanity-check my own understanding here]

[contravariant]

In fact it is enough for functions to be complex differentiable once (in which case they'll automatically be differentiable infinitely often).

[ithinkso]

It's one of those theorems that still boggles my mind even though I know it for so many years now. Either complex functions are so much well behaved or complex differentiability is so much stronger condition, I can't decide which. Top it off with the uniqueness of analytic continuation and you start to wonder what causes real functions to be such a pain.

If anyone knows some nice articles about this topic I would love to read them

[joppy]

One way of understanding why complex differentiability is so strong is looking at a complex-to-complex function as a real function of two real inputs and two real outputs. The fact that h rather than |h| appears in the denominator of the complex derivative causes the derivative to be “aware” of the rotational nature of complex functions: this turns into a differential equation which must be satisfied by the real function (the Cauchy-Riemann equations).

[nimish]

It's the latter. Complex differentiability is a very strong condition.

[ngoldbaum]

Yes, all locally smooth complex functions (e.g. whose real and imaginary parts satisfy the cauchy-riemann equations) are analytic, one of the main reasons complex analysis is way less of a pain than real analysis.