Properties of analytic functions

Question

I read some chapters of the book "Road to Reality".
It seems that analytic functions are one of the more important concepts of the book.
They are listed many properties:

Analytic functions are of class $C^omega$. Why symbol $omega$ ?

Analytic functions are more smooth than $C^infty$ functions. What does it mean ? Only that there are $C^infty$ functions not analytic ? Or there is a more deep meaning ?

An analytic function is really a single function, we can't construct an analytic function gluing together different pieces. I don't understand this one. Where can I find the theorems ?

user3482749 · Answer

1) $omega$ is the ordinal number that is the order type of the natural numbers. I'm not sure if there's a particularly good reason for using it, other than that it's a convenient "infinity" symbol that isn't $infty$ (which is already taken)

2) Mostly just that all analytic functions are $C^infty$, but not all $C^infty$ functions are analytic. There's nothing overly interesting going on there.

3) Probably the Identity Theorem is what you're after here: broadly, if you know what an analytic function does "somewhere" (for a suitable definition of "somewhere"), then you know what it looks like everywhere else. In particular, you can't take two analytic functions on different domains and "join them up" to get an analytic function on all of $mathbb{C}$, except for at most one choice of the second function (in contrast to smooth functions, for which you can always do this if there's a gap between the domains, regardless of what the functions are).

Properties of analytic functions

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