In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit.
Let {fk} be a sequence of holomorphic functions on a connected open set G that converge uniformly on compact subsets of G to a holomorphic function f which is not constantly zero on G. If f has a zero of order m at z0 then for every small enough ρ > 0 and for sufficiently large k ∈ N (depending on ρ), fk has precisely m zeroes in the disk defined by |z − z0| < ρ, including multiplicity.
Indeed, if one chooses a disk such that f has zeroes on its boundary, the theorem fails.
An explicit example is to consider the unit disk D and the sequence defined by which converges uniformly to f(z) = z − 1.
Hurwitz's theorem is used in the proof of the Riemann mapping theorem,[2] and also has the following two corollaries as an immediate consequence: Let f be an analytic function on an open subset of the complex plane with a zero of order m at z0, and suppose that {fn} is a sequence of functions converging uniformly on compact subsets to f. Fix some ρ > 0 such that f(z) ≠ 0 in 0 < |z − z0| ≤ ρ.