I have been studying Conformal field theory for the past one week from the books by Blumenhagen and Di Francesco etal.
If I understand correctly, whenever one talks of 'local (infinitesimal) conformal transformations' on the complex plane (for eg. $ z \rightarrow z + \epsilon(z) $ ) being described by holomorphic functions, one implicitly assumes that there is an open subset (say $U$) containing a point of the Complex plane on which the function $\epsilon(z)$ is well-defined and in fact holomorphic. My first question is whether this is the right way to look at local conformal transformations. If not, then, what exactly does one mean by 'local' conformal transformations?
Now, if the answer to the earlier question is yes, then I don't see why both the above books expand $\epsilon(z)$ as a Laurent series about 0 when deriving the Witt algebra. The Laurent series will have its own domain of convergence which may not contain the open set $U$. So can anybody explain why it is justified to assume that $\epsilon$ has a Laurent series about 0 and use it to derive the form of the generators of the Witt algebra? One could in principle choose any point (say $a$) of the open set and choose the neighbourhood $U$ to be sufficiently 'small' so that $\epsilon(z)$ is represented by a Taylor Series around $a$. One will in general get a smaller algebra by this procedure.
And I repeat: if the answer to my first question is no, could anybody then clarify the meaning of local conformal transformations in the first place?