Inside Collection (Textbook): Analysis of Functions of a Single Variable

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.

Suppose

Though this theorem sounds impressive and perhaps unexpected, it is
really just a combination of (Reference) and the Cauchy Integral Formula.
Indeed, let

Hence, by part (a) of (Reference),

- « Previous module in collection The Open Mapping Theorem and the Inverse Function Theorem
- Collection home: Analysis of Functions of a Single Variable
- Next module in collection » Isolated Singularities, and the Residue Theorem