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0 pointscontravariant5y ago0 comments

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

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ithinkso5y ago

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

joppy5y ago

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).

nimish5y ago

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

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