Analysis I: Lecture 18

In which we define the standard trigonometric and hyperbolic functions and check their properties.

  • Lemma 41 The power series \displaystyle \sum\limits_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!} and \displaystyle \sum\limits_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!} have infinite radius of convergence.  The proof is an exercise (e.g. using the ratio test).
  • Definition of functions s : \mathbb{C} \to \mathbb{C} and c : \mathbb{C} \to \mathbb{C}.
  • Theorem 42
    1. The functions s : \mathbb{C} \to \mathbb{C} and c : \mathbb{C} \to \mathbb{C} are everywhere differentiable, with s'(z) = c(z) and c'(z) = -s(z) for all z \in \mathbb{C}.
    2. We have s(-z) = -s(z) and c(-z) = c(z) for all z \in \mathbb{C}.
    3. We have \displaystyle s(z) = \frac{e^{iz} - e^{-iz}}{2i} and \displaystyle c(z) = \frac{e^{iz} + e^{-iz}}{2} for all z, w \in \mathbb{C}.
    4. For all z, w \in \mathbb{C}, we have s(z+w) = s(z) c(w) + c(z) s(w), and c(z+w) = c(z) c(w) - s(z) s(w), and s(z)^2 + c(z)^2 = 1.

    These were all pretty straightforward from the definitions, using Theorem 34 and various properties of the exponential function.

  • Theorem 43 Consider c : \mathbb{R} \to \mathbb{R} and s : \mathbb{R} \to \mathbb{R}.
    1. If c(a) = c(b) = 0, then s(b-a) = 0.
    2. We have s(x) > 0 for all x \in (0,1].
    3. There is a unique \omega \in [0,2] such that c(\omega) = 0.
    4. We have s(\omega) = 1 for this \omega.

    We used the fact that the error in truncating an alternating series is at most the first term omitted.

  • Definition of \pi = 2\omega.
  • Corollary 44 For z \in \mathbb{C}, we have s(z + 2\pi) = s(z) and c(z + 2\pi) = c(z).
  • Definition of the sine, cosine, hyperbolic sine and hyperbolic cosine functions.

Understanding today’s lecture

Can you prove Lemma 41?

Can you complete the remaining parts of the proof of Theorem 42(iv)?

Can you give an alternative proof of Theorem 42(iv) using the constant value theorem (Lemma 35), in a similar style to Lemma 36?

We proved the addition formulae for sine and cosine for complex numbers (not just for reals).  What do you get if you expand \sin(x+iy) using the addition formula?  You might want to use properties of sine, cosine, and their hyperbolic counterparts.  You could try expanding it using the exponential form of \sin z too, to compare the approaches.

What other facts can you derive about the functions s and c in the style of Corollary 44, using addition formulae and facts about \omega?

Can you write a list of properties of hyperbolic functions (perhaps in the style of Theorem 42) and then prove it from the definitions?

We have just reached the end of a section of the course, so this would be an ideal time to look back over your notes to review what we’ve done so far (and perhaps to update your functions grid), before we move on to something new on Friday.

Further reading

Perhaps some of you are worried that we haven’t mentioned that these various functions that we’re defining are well defined.  That’s because we don’t need to, but if you’re a bit puzzled by what ‘well defined’ means then you might like to read this piece by Tim Gowers.

There’s an interesting description of the history of the trigonometric functions on MacTutor.  They also have an article about the history of the concept of a function, which is a concept that we’ve been using rather a lot in this course.

Preparation for Lecture 19

How would you rigorously define integration?  What properties do you expect the integral to have?  Do you expect to be able to integrate every function?  Can you give examples of integrable and non-integrable functions?

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2 Responses to “Analysis I: Lecture 18”

  1. anonymous Says:

    Hi Vicky

    I was just wondering if you could point us in the direction of any articles you know about turning our analysis definitions of the trig functions into geometrical ones?

    Thanks

  2. theoremoftheweek Says:

    I’m sure that there are lots of references for this, so if you know of any good books or online articles that discuss it then please do mention them here.

    Professor Körner put a couple of relevant comments/results at the end of section 10 of his lecture notes from the Analysis I course a few years ago. (Note that his printed notes do not contain proofs, so you’ll need to supply those yourself!)

    He has written more about this in his Companion to Analysis book: there’s quite a bit about the link between analysis and geometry via trig functions, so if you’re interested then that would be an ideal place to start looking.

    There’s also a discussion here that might help (but it doesn’t go into as much depth as the book).

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