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<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>.


Analytic continuation

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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.


Holomorphic function

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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.


Ciro Santilli @cirosantilli 37

 Incoming links: Identity theorem