# Notes - Complex Analysis MT23, Power series

### Flashcards

- Standard power series results from
[[Notes - Analysis I MT22, Power series]]
^{U}and [[Notes - Analysis III TT23, Power series]]^{U}.

Suppose $\sum _ {n = 0}^\infty a _ n z^n$ is a power series, $S$ the subset of $\mathbb C$ on which is converges and $R$ its radius of convergence. What containment do we have in terms of balls about $0$, and what can you say about when the series converges absolutely and uniformly?

\[B(0, R) \subseteq S \subseteq \bar B(0, R)\]
The series converges absolutely on $B(0, R)$ and if $0 \le r < R$ then it converges uniformly on $\bar B(0, r)$.

Suppose $\sum _ {n = 0}^\infty a _ n z^n$ is a power series, $S$ the subset of $\mathbb C$ on which is converges and $R$ its radius of convergence. Can you give an explicit formula for its radius of convergence?

Suppose $\sum _ {n=0}^\infty a _ n z^n$ and $\sum^\infty _ {n = 0} b _ n z^n$ are power series with radius of convergence $R _ 1$ and $R _ 2$ respectively. What can you say about the sum and product of these power series?

- $\sum^\infty _ {n = 0} (a _ n + b _ n) z^n$ converges if (but not only if) $ \vert z \vert < \min(R _ 1, R _ 2)$.
- $(\sum _ {n=0}^\infty a _ n z^n)(\sum^\infty _ {n = 0} b _ n z^n) = \sum^\infty _ {n = 0} (\sum _ {k+l = n} a _ k b _ l) z^n$ converges if (but not only if) $ \vert z \vert < \min(R _ 1, R _ 2)$.

How are $\sin(z)$ and $\cos(z)$ defined in terms of a power series?

### Proofs

Prove the following:

Let $\sum _ {n = 0} a _ n z^n$ be a power series, let $S$ be the subset of $\mathbb C$ on which is converges and let $R$ be its radius of convergence. Then

\[B(0, R) \subseteq S \subseteq \bar B(0, R)\]
The series converges absolutely on $B(0, R)$ and if $0 \le r < R$ then it converges uniformly on $\bar B(0, r)$. Moreover, we have

\[\frac 1 R = \limsup_n |a_n|^{1/n}\]

Todo.