Complex analysis

The surprising thing is that a bunch of results are simpler in complex analysis!

Being a complex holomorphic function is an extremely strong condition.

The existence of the first derivative implies the existence of all derivatives.

Another extremely strong consequence is the identity theorem.

"Holos" means "entire" in Greek, so maybe this is a reference to the fact that due to the identity theorem, knowing the function on a small open ball implies knowing the function everywhere.

visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.

The key question is: how can this continuation be unique since we are defining the function outside of its original domain?

The answer is: due to the identity theorem.

Good ultra quick visual non-mathematical introduction to the Riemann hypothesis and analytic continuation.

Essentially, defining an holomorphic function on any open subset, no matter how small, also uniquely defines it everywhere.

This is basically why it makes sense to talk about analytic continuation at all.

One way to think about this is because the Taylor series matches the exact value of an holomorphic function no matter how large the difference from the starting point.

Therefore a holomorphic function basically only contains as much information as a countable sequence of numbers.

visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.

One of the Millennium Prize Problems and Hilbert's problems.