3 ms·
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
by ithinkso 6y 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
- nimish 6y agoIt's the latter. Complex differentiability is a very strong condition.
- joppy 6y agoOne 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).