Area: Analytic Continuation in Complex Analysis
Question Does the shape of the domain matter when doing analytic continuation along a path?
Background I have been reading up on the concept of analytic continuation and have realized that some textbooks introduce the concept of analytic continuation along a path with the help of disks (see image below)
but other textbooks (for example Needham (1997) page 251) seem to let the domains connecting the points look like anything.
I guess that some authors like to introduce the concept with the help of power series representation of analytic functions and therefore using disks for the continuation but this seems cheap in my opinion if one could introduce it like Needham and letting the reader know that the domains can look like anything and are not confined to only be disks.
But I start to doubt myself when I look at wikipedia and other theorems connected to this area since for example Monodromy theorem uses disks as the shapes on Wikipedia (https://en.wikipedia.org/wiki/Monodromy_theorem).
Anyone can let me know if the shape of the domain matters when doing analytic continuation along a path?
Best Answer
Briefly, it doesn't matter if we perform analytic continuation using disks or general connected open sets: The resulting notions of analytic continuation along a path coincide.
Every connected open set is path-connected, and may be written as a union of open disks. In a connected open set, analytic continuation along a path using open disks is unique.
If instead we perform analytic continuation along a path using arbitrary connected open sets, the preceding paragraph reduces us to using disks.
Restricting to disks may be technically and conceptually simpler (sometimes less is more even when it comes to choices), and disks are the natural regions of convergence of power series.