importance of Unit ball in complex analysis

Question

I am trying to study some papers of Krantz on holomorphic functions and important spaces on the unit ball such as Bergman and specially Bloch spaces and suddenly it occurred to me why we study functions on unit ball? I feel like there is more than because It is a simpler case! I tried to find the reason but all books and papers around are so technical I need to see the big picture! Kehe Zhu's book and Krantz's book haven't been a great help. any reference introduction would be appreciated. thanks in advance.

Answer 1

My opinion.

The unit ball is diffeomorphic to every other ball in the complex plane. If a holomorphic function is defined on some area, then, for a sufficiently large radius, there is a ball that encompasses that area. Moreover, since holomorphic functions are generally defined on open domains - to even make differentiability make sense - about any point in a function's (open) domain, there is a sufficiently small ball about this point over which the function is defined. So, one may as well study the unit ball for notational convenience since, without too much loss of generality, you can apply what you've learnt to many other situations. The only thing I can think of right now that doesn’t seem to transfer is the property that $\{z^n\}$ has good convergence properties in the unit ball, and diverges outside of it. But maybe you’d be considering $(z-z_0)^n$ anyway, in a different setting.

Thanks to the identity theorem, a holomorphic function is also, in some sense, determined by its action over the unit ball.

Think also about power series - which every holomorphic function has - where we have summands $c_n(z-z_0)^n$. Does it matter what $z_0$ is? Not really. If I study power series of the form $c_nz^n$ (as I would, working in the unit ball) then essentially everything transfers painlessly to power series of the form $c_n(z-z_0)^n$. All that has happened is a potential translation of the function by $z_0$. This will not affect properties of interest!

This theme plays out in topology too. When I was studying some functional analysis, I noted that my textbook always discussed the (weak)(weak-star)(strong)-compactness of the unit ball, or lack thereof. Why? Because we could get results about any other closed and bounded set immediately, because such sets are closed subsets of a ball of larger radius - but all balls, of any radius, are homeomorphic, so these balls were also compact (and hence their closed subsets were compact).