Transforming imaginary and complex numbers

Difference equations are the topic of interest in Chapter 1 of Hamilton's Time Series Analysis. The chapter includes a section that outlines the general solution of pth-order difference equations with distinct eigenvalues. On page 14, Hamilton says that:

interesting possibilities arise if some of the eigenvalues are complex

and then a series of equations follow this statement. To guide the reader through the equations, footnotes are provided at the bottom of the page that refer the reader to the mathematical appendix at the back of the book.

It can be a little bit difficult to figure out what's going on in the equations on pages 14 to 16 of the book, so I thought it would be a good idea to write up some notes in order to make the material more understandable.

The equations involve transforming imaginary and complex numbers to trigonometric functions. It's straightforward stuff really, but since no reference is made to the names of the formulas being used, it may leave you scratching your head as to what's going on.

So, our aim here is to understand how to transform imaginary and complex numbers to trigonometric functions. To do this, we're going to need to familiarize ourselves with the following:

1. Expressing Cartesian coordinates in polar coordinate form; 
2. Euler's formula; 
3. De Moivre's Theorem.

Let's begin with expressing Cartesian coordinates in polar coordinate form.

$a$ and $b$ in the diagram below are called Cartesian coordinates. These can be expressed in terms of $\theta$ and $R = \sqrt{a^{2}+b^{2}}$ which are called polar coordinates. To do this, we use the following formulas: \[a = R~cos~\theta,~ ~~~~~ b = R~sin~\theta,~ ~~~~~ R>0.\] Applying these formulas to the conjugate complex number $(a \pm b i)$, we have: \[a \pm b i = R~cos~\theta \pm i~R~sin~\theta = R(cos~\theta \pm i~sin~\theta) \] and this explains Equation [1.2.36] in Hamilton.

Second on the list is Euler's formula. This tells us that: \[ e^{\pm i \theta} = cos~\theta \pm i~sin~\theta.\] Thus we can express $(a \pm bi) = R(cos~\theta \pm i~sin~\theta)$ as \[a \pm bi = R(cos~\theta \pm i~sin~\theta) = R e^{\pm i \theta}.\] The last item on the list is De Moivre's theorem, but first let's see how a conjugate complex number is raised to the jth power. It is done as follows: \[(a \pm bi)^{j} = (Re^{\pm i \theta})^{j} = R^{j} e^{\pm i j \theta}.\] Or, using De Moivre's theorem we have: \[(a \pm bi)^{j} = R^{j}(cos~j~\theta \pm i~sin~j~\theta).\] and now we've arrived at Equation [1.2.37] in Hamilton.

Although not explicitly mentioned in the textbook, it's Euler's formula and De Moivre's theorem that lead to both Equation [1.2.37] and Equation [1.2.38] in Chapter 1 of Hamilton.

The rest of the section on the general solution of pth-order difference equations with distinct eigenvalues involves no more than basic substitution and a little bit of algebra. Despite the fact that the notation looks a bit "heavy", it's easy to follow once you know to use Euler's formula and De Moivre's theorem.

Vector Algebra (complete proofs)

Open up a decent undergraduate Linear Algebra textbook (my favourites are Fraleigh & Beauregard and Strang) and you'll quickly come across definitions of vector addition, vector subtraction, and scalar multiplication. A list of the properties of these vector operations usually follow and while one or two proofs are given, the rest are left up to the reader as exercises. That shouldn't be too much of a problem to any mathematics student.

You will, however, encounter such proofs - again left as exercises - in graduate textbooks for economics and finance; one such example being Harrison & Waldron. They also appear in this Advanced Engineering Mathematics textbook that I happened to pick up at a bargain price in Oxfam!

Familiarity with vector algebra is pretty essential to anyone studying a numerical subject and up to a certain level students can advance in their chosen field without being able to write proofs.

There comes a time, though, when it is necessary for the student to be able to understand and write proofs. That's got to be a nightmare for anyone who has not yet grasped how to do it. Even worse, there's probably no course designed to teach it to someone for the first time. Gone are the days of the trivium.

Mathematics students at Virginia Commonwealth University are quite the lucky ones because they have a whole course designed to teach them how to write mathematical proofs. The course is called Introduction to Mathematical Reasoning and the book that's used for the course can be found here (it's an excellent textbook!).

Students of economics are, perhaps, not so lucky. Some may already lack math skills (or rather, the confidence in their math ability), and I've not really come across any material that aims to teach them how to write proofs. It's kind of like they're left to their own devices to try and work it out for themselves. A tad unfair - remember, some 3rd year math students take a whole course in this art!

Now, to write proofs for the properties of vector algebra is very easy and as basic as you can get, but since someone out there (maybe a student of economics) might be having trouble with writing them, I've written up complete proofs and added comments to them that explain how each step in the proof was made. I think they're quite instructive and definitely more detailed than any I've seen in a variety of textbooks. Feedback welcome.

Note: proofs can be written for the properties of matrix algebra in a similar fashion. Following the tradition, I'll leave them as an exercise. Follow the technique I used for vector algebra and you'll have no problem.