a good book is hille's analytic function theory, or cartan's analytic functions of one and several complex variables. also george mackey's harvard notes on complex variables. I think I know something about it too, but I don't feel like writing you a text right now. if you ask a question i will try to answer.
the basic question is to take a complex power series converging in a disc, and consider a connected open set U containing that disc, and then ask for an extension into U, of the analytic function defined in the disc by the power series. it may or may not exist and may or may not be unique.
If the connected set U is also "simply connected" which means it contains no loops that encircle points outside the set, then there is at most one extension to the whole set U.
as a "counterexample" to uniqueness, the complement U of the origin is not simply connected, since it contains a loop, namely the unit circle that encircles the origin, which is outside the set. The power series for ln(z) which converges in the unit disc centered at z=1, has no analytic extension into the entire set U, but has exactly one such extension into any set of form "U minus a ray extending from the origin to infinity".
If U is a connected open set, and if there exists a (unique) analytic extension of your function into every simply connected (open) subset of U, then in fact there is a maximal analytic extension, not into any subset of the plane, but rather into a simply connected "covering" of the set U.
This topic, Riemann surfaces, was the subject of a groundbreaking book by the famous Hermann Weyl, also known for his work on group representations.
I hope I have not said anything false.
