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The Fundamental Theorem of Algebra, Analysis: Uniform Convergence of Analytic Functions

Module by: Lawrence Baggett. E-mail the author

Summary: A brief module containing a theorem about uniform convergence of analytic functions.

Part (c) of (Reference) gives an example showing that the uniform limit of a sequence of differentiable functions of a real variable need not be differentiable. Indeed, when thinking about uniform convergence of functions, the fundamental result to remember is that the uniform limit of continuous functions is continuous ((Reference)). The functions in (Reference) were differentiable functions of a real variable. The fact is that, for functions of a complex variable, things are as usual much more simple. The following theorem is yet another masterpiece of Weierstrass.

Theorem 1

Suppose UU is an open subset of C,C, and that {fn}{fn} is a sequence of analytic functions on UU that converges uniformly to a function f.f. Then ff is analytic on U.U. That is, the uniform limit of differentiable functions on an open set UU in the complex plane is also differentiable on U.U.

Proof

Though this theorem sounds impressive and perhaps unexpected, it is really just a combination of (Reference) and the Cauchy Integral Formula. Indeed, let cc be a point in U,U, and let r>0r>0 be such that B¯r(c)⊆U.B¯r(c)⊆U. Then the sequence {fn}{fn} converges uniformly to ff on the boundary CrCr of this closed disk. Moreover, for any z∈Br(c),z∈Br(c), the sequence {fn(ζ)/(ζ-z)}{fn(ζ)/(ζ-z)} converges uniformly to f(ζ)/(ζ-z)f(ζ)/(ζ-z) on Cr.Cr. Hence, by (Reference), we have

f ( z ) = lim f n ( z ) = lim n 1 2 π i ∫ C r f n ( ζ ) ζ - z d ζ = 1 2 π i ∫ C r f ( ζ ) ζ - z d ζ . f ( z ) = lim f n ( z ) = lim n 1 2 π i ∫ C r f n ( ζ ) ζ - z d ζ = 1 2 π i ∫ C r f ( ζ ) ζ - z d ζ .
(1)

Hence, by part (a) of (Reference), ff is expandable in a Taylor series around c,c, i.e., ff is analytic on U.U.

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