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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.




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


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


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


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


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).


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


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.




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