Hurwitz's theorem

Hurwitz's theorem states that a locally uniform limit of zero-free holomorphic functions on a domain is either zero-free or identically zero.

Rouché's theorem: If two holomorphic functions differ on a contour by less than the modulus of one, they have the same number of enclosed zeros. Rouché's theorem turns local uniform convergence into preservation of zero counts near an alleged limit zero.
Open set: A subset of a topological space is open when every point has a neighborhood contained in the subset. The theorem is applied on a domain, an open connected subset of the complex plane.
Adolf Hurwitz: Adolf Hurwitz was a German mathematician known for work in number theory, algebra, and complex analysis. The theorem bears his name and reflects his contributions to classical function theory.
Riemann mapping theorem: Every simply connected proper domain in the complex plane is conformally equivalent to the open unit disk. Hurwitz's theorem helps establish that limits of injective holomorphic maps remain injective or degenerate.
Uniform convergence: A sequence converges uniformly when a single error bound works at every point of its domain. Hurwitz requires uniform convergence only on compact subsets, not necessarily across the entire domain.
Argument principle: A contour integral of the logarithmic derivative counts a holomorphic function's zeros minus poles inside the contour. It explains why a small boundary perturbation cannot change the zero count used in Hurwitz's theorem.
Domain (complex analysis): A domain is a nonempty connected open subset of the complex plane. Connectedness makes the theorem's exceptional alternative, an identically zero limit, meaningful across the whole set.
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work shaped complex analysis and geometry. Hurwitz's function theory developed within the analytic framework established by Riemann.
Univalent function: A univalent function is a holomorphic function that is one-to-one on its domain. Applying Hurwitz to differences of values proves locally uniform limits of univalent functions are univalent or constant.
Pointwise convergence: A sequence of functions converges pointwise when its values converge separately at each input. Pointwise convergence alone does not ensure that zero-freeness survives in the limit.

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