Hurwitz's theorem
/ HOOR-vits /
When a sequence of holomorphic functions settles down to a limit function, what happens to their zeros? Do the roots of the approximations have anything to do with the roots of the limit? Hurwitz's theorem gives a reassuring answer: zeros are preserved in the limit. The roots cannot suddenly appear out of nowhere or vanish into thin air as you pass to the limit, as long as the convergence is uniform on compact sets.
The statement: suppose holomorphic functions f_n converge to f uniformly on every compact subset of a region (locally uniform convergence), and f is not identically zero. Then for any point z_0 where f has a zero, every neighborhood of z_0 eventually contains zeros of f_n: the approximating functions develop zeros that converge to the zero of the limit. More sharply, around a small circle on which f has no zeros, the number of zeros of f_n inside equals the number of zeros of f inside for all large n. The proof is a clean application of Rouche's theorem: on a small circle around z_0 the limit f is bounded away from 0, and uniform convergence makes |f_n - f| < |f| there for large n, so f_n and f have the same zero count inside that circle.
A striking corollary is about injectivity: a locally uniform limit of injective (one-to-one) holomorphic functions is either injective or constant. This is exactly the kind of stability needed to run compactness arguments in geometric function theory — for instance to extract a limit map in the proof of the Riemann mapping theorem and know it is still univalent. The honest caveat: the hypotheses bite. Convergence must be locally uniform (not merely pointwise), and the limit must not be the zero function — for the constant zero limit the conclusion about isolated zeros is meaningless. And the theorem is about where zeros go, not a guarantee that f_n shares the exact multiplicity at the exact point.
The partial sums of e^z = sum z^n / n! converge to e^z uniformly on compact sets. Since e^z is never zero, Hurwitz's theorem says that on any fixed disk the partial-sum polynomials eventually have no zeros there either — their roots must flee toward infinity.
Zeros of the approximations track the zeros of the limit; with no limit zeros, the roots run off to infinity.
It requires locally uniform convergence and a not-identically-zero limit; mere pointwise convergence is not enough, and the corollary about injective limits genuinely allows the constant function as the other alternative.