Analysis I: Lecture 17

In which we define some standard functions and check that they behave as we expect.

• Theorem 37
1. The function $e : \mathbb{R} \to \mathbb{R}$ is differentiable on $\mathbb{R}$, with derivative $e'(x) = e(x)$ for all $x$.
2. $e(x+y) = e(x) e(y)$ for all $x$, $y \in \mathbb{R}$.
3. $e(x) > 0$ for all $x \in \mathbb{R}$.
4. The function $e : \mathbb{R} \to \mathbb{R}$ is strictly increasing.
5. $e(x) \to \infty$ as $x \to \infty$, and $e(x) \to 0$ as $x \to -\infty$.
6. The function $e : \mathbb{R} \to (0, \infty)$ is a bijection.

We proved (i) and (ii) earlier, when we were thinking of $e$ as a function from $\mathbb{C}$ to $\mathbb{C}$.  We then carefully checked the remaining properties using these.  We used the intermediate value theorem for (vi).  We noted that (ii) and (vi) show that the function $e$ is a group isomorphism between the groups $(\mathbb{R},+)$ and $((0,\infty),\times)$.

• Definition of the function $\ell: (0,\infty) \to \mathbb{R}$.
• Theorem 38
1. $\ell: (0, \infty) \to \mathbb{R}$ is a bijection.  We have $\ell(e(x)) = x$ for all $x \in \mathbb{R}$, and $e(\ell(t)) = t$ for all $t \in (0, \infty)$.
2. The function $\ell : (0, \infty) \to \mathbb{R}$ is everywhere differentiable, with $\ell'(t) = \frac{1}{t}$ for $t \in (0, \infty)$.
3. $\ell(tu) = \ell(t) + \ell(u)$ for all $t$, $u \in (0, \infty)$.

This all followed from the fact that $\ell$ is the inverse of $e$.  We used the inverse function theorem for (ii).

• Definition of the function $r_{\alpha} : (0, \infty) \to \mathbb{R}$ for real $\alpha$.
• Theorem 39 Take $x$, $y \in (0, \infty)$ and $\alpha$, $\beta \in \mathbb{R}$.  Then
1. $r_{\alpha}(xy) = r_{\alpha}(x) r_{\alpha}(y)$;
2. $r_{\alpha+\beta}(x) = r_{\alpha}(x) r_{\beta}(x)$;
3. $r_{\alpha}(r_{\beta}(x)) = r_{\alpha \beta}(x)$; and
4. $r_1(x) = x$.

This was straightforward using the definition and the properties of the functions $e$ and $\ell$.

• Lemma 40
1. For $\alpha \in \mathbb{R}$, the function $r_{\alpha} : (0, \infty) \to (0, \infty)$ is everywhere differentiable, and $r_{\alpha}'(x) = \alpha r_{\alpha - 1}(x)$.
2. For $x > 0$, define $f_x : \mathbb{R} \to \mathbb{R}$ by $f_x(\alpha) = x^{\alpha}$.  The function $f_x$ is everywhere differentiable, and $f_x'(\alpha) = \log x f_x(\alpha)$.

We used the chain rule for each part of this.

Understanding today’s lecture

Why did we focus on the restriction of the function $e$ to the real numbers in Theorem 37?

Can you show directly that $\ell(tu) = \ell(t) + \ell(u)$ for all positive real numbers $t$ and $u$ (without using facts from group theory)?

Were you surprised by any of the definitions?  What other definitions might we have chosen?  What was convenient (or inconvenient) about the definitions as described in lectures?

As we’re defining these standard functions in lectures, you could add them (or new functions built from them) to your functions grid to increase your repertoire of useful examples.

I hope that you’re not just taking my word for it when we apply theorems in lectures: you should go back and check that the conditions of the theorem being applied really are satisfied.  For example, today I quoted Corollary 25(i), which officially we stated for functions defined on closed intervals.  Why were we indeed allowed to apply it as we did in today’s lecture?