Intro Complex Analysis, Lec 29, Uniform Convergence, Taylor Series Facts

Lecture 29. (0:30) Quiz 5 due by midnight. (0:58) Exam 3 in one week. (2:44) Plan for the lecture. (3:07) Definition of pointwise convergence of a sequence of complex functions over some subset of the complex plane. (8:48) Uniform convergence in terms of an epsilon-tube (for a real-valued function) and the definition. (15:47) Visualize uniform convergence for complex-valued functions. (20:13) Facts about power series and Taylor Series (Maclaurin series). Also include 1) an example of an infinitely differentiable real-valued function whose Taylor series only converges to it at one point and 2) a long-division-based computation of the Taylor series for tan(z) centered at z = 0. (51:33) Taylor series calculation using tricky algebra and the formula of a convergent geometric series.

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.

[conorbrennan]

Well Billy, why can't we just look at pointwise convergence of a sequence of analytic functions on an open disc to guarantee the limit function is analytic, I can't find any counter examples so it must be true, uniform convergence probably too strong.