## Analysis I: Lecture 16

In which we study the differentiability of power series and define a new function.

• Lemma 33
1. For $2 \leq r \leq n$, we have $\binom{n}{r} \leq n(n-1)\binom{n-2}{r-2}$.
2. For $h$, $z \in \mathbb{C}$, we have $| (z+h)^n - z^n - nhz^{n-1}| \leq n(n-1)(|z| + |h|)^{n-2}|h|^2$.

These were both straightforward estimates.

• Theorem 34 Let $\sum\limits_{n=0}^{\infty} a_n z^n$ be a complex power series with radius of convergence $R$, so that we may define $f(z) = \sum\limits_{n=0}^{\infty} a_n z^n$ for $z$ with $|z| < R$.  Then $f$ is differentiable on $\{ z \in \mathbb{C} : |z| < R \}$, and $f'(z) = \sum\limits_{n=1}^{\infty} n a_n z^{n-1}$.
• Definition of a function $e:\mathbb{C} \to \mathbb{C}$ by $e(z) = \sum\limits_{n=0}^{\infty} \frac{z^n}{n!}$.
• Lemma 35 (Constant value theorem) Let $f : \mathbb{C} \to \mathbb{C}$ be a differentiable function.  If $f'(z) = 0$ for all $z \in \mathbb{C}$, then $f$ is constant.  We proved this using the constant value theorem for real functions (Corollary 25(iii)).
• Lemma 36 We have $e(z+w) = e(z) e(w)$ for all $z$, $w \in \mathbb{C}$.  We proved this by applying the Constant value theorem to a suitable function.

#### Understanding today’s lecture

What was it about the auxiliary function in Lemma 36 that made it work?  What properties did we want it to have?  Can you see why I chose to use that particular function?

How else might we have defined the function $e : \mathbb{C} \to \mathbb{C}$ (secretly thinking of it as the exponential function)?  If you choose a different definition, can you derive the properties of the exponential function from that definition?

You could add the function $e$ to your functions grid.  Which properties does it have?

When you have learned a bit more about complex analysis, you will find that there are other ways to prove Lemma 35.  Here’s a page with a couple of approaches.

You might be interested in this piece by Tim Gowers about what we mean by a definition.

#### Preparation for Lecture 17

What further properties do we expect the exponential function to have?  Can you prove them from our definition?

What properties do we expect to define the logarithm function to have?  How might we define the logarithm function and then derive its properties?

How might we define the trigonometric functions?  How might we then derive their properties?