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?

Further reading

Inevitably Wikipedia has something to say about the exponential function and the logarithm function.

A curious feature of the widespread availability of pocket calculators was the change in the order in which school students encountered various key ideas.  Prior to the calculator, students learned about exponentials and logarithms because they enabled one to carry out computations that would otherwise be impractical (for example, with the help of a slide rule), and they then encountered calculus later.  Now that we all have easy access to devices that can do large calculations for us, we typically encounter exponentials and logarithms after we have learned about differentiation, and often motivate their study using calculus.  Which properties seem important depends entirely on context and other knowledge.  Just a little historical footnote.

Preparation for Lecture 18

How might we define the trigonometric and hyperbolic functions?  What properties do we want them to have?  Can you prove that they have these properties?


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