• Complex Analysis: #13 Weierstrass`s Convergence Theorem

This is the analog of the theorem in real analysis which states that a sequence of continuous functions which is uniformly convergent converges to a continuous function. But here we are concerned with analytic functions.


Theorem 28 (Weierstrass)
Let G₁ ⊂ G₂ ⊂ · · · ⊂ G_n ⊂ · · · be an increasing sequence of regions in ℂ and let (f_n)_n∈ℕ be a sequence of analytic functions f_n : G_n → ℂ for each n. Let G = ∪_n=1^∞ G_n (that is, n = 1, . . ., ∞), and assume that in every compact subset of G, the sequence (f_n) converges uniformly. Let f : G → ℂ be defined by f(z) = lim_n→∞ f_n(z) for all z ∈ G. Then f is analytic on G, and furthermore f_n' → f ' uniformly on every compact subset of G.

Proof
Let z₀ ∈ G and take r > 0 such that B(z₀, r) ⊂ G. (That is the closure of the open set B(z₀, r).) Let N ∈ ℕ be sufficiently large that B(z₀, r) ⊂ ∪_n=1^N G_n. Then B(z₀, r) ⊂ G_m, for all m ≥ N. According to theorem 2 we have ∫_γf_n(z)dz = 0 for all triangles in B(z₀, r). Moreover we have

owing to the uniform convergence of the sequence. Therefore, by theorem 10, f is analytic. By Goursat’s theorem, the derivatives are also analytic. Specifically, let ζ be the boundary of the disc B(z₀, r). Then, using Cauchy’s formula (theorem 6), we have

and the fact that the convergence of the f_n is uniform in B(z₀, r) shows that f_n' → f ' uniformly.

An interesting example of this is Riemann’s Zeta function. Let z = x + iy with x > 1. Then

Thus the infinte sum ζ(z) = ∑ n^−z  (with n =1, . . . , ∞) defines a function which is the limit of a uniformly convergent sequence of analytic functions (the partial sums) for all z with Re(z) > a, where a > 1 is a given constant. Therefore Weierstrass’s convergence theorem implies that the Zeta function ζ(z) is analytic in this region. As a matter of fact, Riemann showed that, with the exception of the obvious isolated singularity at the point z = 1, the zeta function can be analytically continued throughout the whole complex plane. The big question is “Where are the zeros of the zeta function?” Some of them are located at negative even integers. (These are the so-called “trivial zeros”.) They are not particularly interesting. But there are lots along the vertical line Re(z) = 1/2. The famous Riemann conjecture — which is certainly the greatest unsolved problem in mathematics today — is that all of the zeros (apart from the trivial ones) lie on this line. [An American businessman, Mr. Landon T. Clay has put aside one million American dollars each for the solution of a certain collection of outstanding problems in mathematics. The Riemann conjecture is one of them.]