## Analysis I: Lecture 14

In which we think about where a complex power series converges.

• Lemma 30 Let $\sum\limits_{n=0}^{\infty} a_n w^n$ be a convergent series (with $w \neq 0$.)  If $|z| < |w|$, then $\sum\limits_{n=0}^{\infty} a_n z^n$ converges absolutely, and so converges.  We proved this by comparing the series with the convergent geometric series $\sum\limits_{n=0}^{\infty} \left|\frac{z}{w}\right|^n$.
• Theorem 31 (Existence of a radius of convergence) Let $\sum\limits_{n=0}^{\infty} a_n z^n$ be a complex power series.  Then either the series converges absolutely for all $z \in \mathbb{C}$, or there is a non-negative real number $R$ such that the series converges absolutely if $|z| < R$ and diverges if $|z| > R$We’ll prove this next time, using Lemma 30 repeatedly.

#### Understanding today’s lecture

The summary above suggests that we didn’t do much, but that’s because we spent quite a lot of the lecture thinking about various important examples.

To check your understanding of complex differentiability, you could try some more examples of functions.  Can you find where they are differentiable?  What’s the most exotic function that you can study?

You could experiment with some examples of power series.  Given a series, we’re interested in the set of points at which it converges.  Can you go the other way?  That is, given a set in the complex plane, can you find a power series that converges at points in the set and diverges at points not in the set?  Which sets work for this and which do not?

As a sneak preview of Complex Analysis or Complex Methods next year, you could read about the link between differentiable functions and power series in $\mathbb{C}$, Liouville’s theorem, and Cauchy’s integral theorem, for example.  There are good suggestions for books on this subject in the Schedules for Complex Analysis and Complex Methods.

Gareth Taylor (not, as far as I know, any relation of Brook Taylor) has some notes on Taylor’s theorem.

Tim Gowers wrote some blog posts about past Tripos questions.  At the end of one of those posts, he wrote about what is for us Lemma 30.

#### Preparation for Lecture 15

Can you prove Theorem 30?

What can you say about when we can differentiate a power series by differentiating each term individually (‘term-by-term differentiation’)?