Re: Understanding exercise 9.3

vasco wrote:

So what is this 9.3 exercise all about, since it seems to me that F and G are already holomorphic on the Riemann sphere?

Not quite. As you said yourself, in general F is only holomorphic in a neighbourhood of the origin and G is only holomorphic in a neighbourhood of complex infinity, so neither need be holomorphic on the entire sphere. If these neighbourhoods overlap then there is an annulus where both functions are simultaneously holomorphic and thus where F+G is defined.

The statement of 9.3 can be elaborated as follows. Let F be any function defined on the unit circle. Show that there exist functions F_+, F_- defined uniquely up to a constant term, such that F = F_+ + F_- and where F_+ is holomorphic in a neighbourhood of complex infinity and F_- is holomorphic in a neighbourhood of the origin.

Does this make sense?