Abstract: We will start by showing that generically for all holomorphic functions f on a planar simply connected domain every order's antiderivative is unbounded. This leads to consider multivalued antiderivatives of any order on every non simply connected domain. If on such a domain V a holomorphic function f has a one-valued antiderivative of any order in V, then f has a holomorphic extension on the simply connected envelop of V. On a non simply connected domain generically every holomorphic function deos not have a one-valued primitive.

Can the space H(V) of all holomorphic functions be replaced in the above results by other spaces of holomorphic functions?