Appendix A Review of complex numbers
Let’s quickly review some basic facts about complex numbers that are typically covered in an earlier course. First, we define the set of complex numbers by
\begin{equation*}
\C = \{x+iy \,|\, x,y\in \R\},
\end{equation*}
where \(i=\sqrt{-1}\text{.}\) We have a bijection \(\C \to \R^2\) given by \(x+iy\mapsto (x,y)\text{;}\) because of this, we often picture \(\C\) as the complex plane, with a “real” \(x\) axis, and an “imaginary” \(y\) axis.
Arithmetic with complex numbers is defined by
\begin{align*}
(x_1\amp +iy_1)+(x_2+iy_2) = (x_1+x_2)+i(y_1+y_2) \\
(x_1\amp +iy_1)(x_2+iy_2) = (x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1)\text{.}
\end{align*}
The multiplication rule looks complicated, but it’s really just “FOIL”, along with the fact that \(i^2=-1\text{.}\) Note that if \(c=c+i0\) is real, we have \(c(x+iy)=(cx)+i(cy)\text{,}\) so that \(\C\) has the structure of a two dimensional vector space over \(\R\) (isomorphic to \(\R^2\)).
Subtraction is defined in the obvious way. Division is less obvious. To define division, it helps to first introduce the complex conjugate. Given a complex number \(z=x+iy\text{,}\) we define \(\overline{z}=x-iy\text{.}\) The importance of the conjugate is that we have the identity
\begin{equation*}
z\bz = (x+iy)(x-iy)=x^2+y^2\text{.}
\end{equation*}
So \(z\bz\) is real, and non-negative. This lets us define the modulus of \(z\) by
\begin{equation*}
\abs{z} = \sqrt{z\bz} = \sqrt{x^2+y^2}\text{.}
\end{equation*}
This gives a measure of the magnitude of a complex number, in the same way as the vector norm on \(\R^2\text{.}\)
Now, given \(z=x+iy\) and \(w=s+it\text{,}\) we have
\begin{equation*}
\frac{z}{w}=\frac{z\bar{w}}{w\bar{w}} = \frac{(x+iy)(s-it)}{s^2+t^2} = \frac{xs-yt}{s^2+t^2}+i\frac{xt+ys}{s^2+t^2}\text{.}
\end{equation*}
And of course, we have \(w\bar{w}\neq 0\) unless \(w=0\text{,}\) and as usual, we don’t divide by zero.
An important thing to keep in mind when working with complex numbers is that they follow the same algebraic rules as real numbers. For example, given \(a,b,z,w\) all complex, and \(a\neq 0\text{,}\) where \(az+b=w\text{,}\) if we want to solve for \(z\text{,}\) the answer is \(z=\frac1a(w-b)\text{,}\) as it would be in \(\R\text{.}\) The difference between \(\R\) and \(\C\) only really materializes when we want to compute \(z\text{,}\) by plugging in values for \(a,b\) and \(w\text{.}\)
One place where \(\C\) is computationally more complicated is finding powers and roots. For this, it is often more convenient to write our complex numbers in polar form. The key to the polar form for complex numbers is Euler’s identity. For a unit complex number \(z\) (that is with \(\abs{z}=1\)), we can think of \(z\) as a point on the unit circle, and write
\begin{equation*}
z = \cos(\theta)+i\sin(\theta)\text{.}
\end{equation*}
If \(\abs{z}=r\text{,}\) we simply change the radius of our circle, so in general, \(z = r(\cos(\theta)+i\sin(\theta))\text{.}\) Euler’s identity states that
\begin{equation}
\cos(\theta)+i\sin(\theta)=e^{i\theta}\text{.}\tag{A.0.1}
\end{equation}
This idea of putting a complex number in an exponential function seems odd at first. If you take a course in complex variables, you’ll get a better understanding of why this makes sense. But for now, we can take it as a convenient piece of notation. The reason it’s convenient is that the rules for complex arithmetic turn out to align quite nicely with properties of the exponential function. For example, de Moivre’s Theorem states that
\begin{equation*}
(\cos(\theta)+i\sin(\theta))^n = \cos(n\theta)+i\sin(n\theta)\text{.}
\end{equation*}
This can be proved by induction (and the proof is not even all that bad), but it seems perfectly obvious in exponential notation:
\begin{equation*}
(e^{i\theta})^n = e^{in\theta}\text{,}
\end{equation*}
since you multiply exponents when you raise a power to a power.
Similarly, if we want to multiply two unit complex numbers, we have
\begin{align*}
(\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta) \amp = (\cos\alpha\cos\beta-\sin\alpha\sin\beta)\\
\amp \quad\quad +i(\sin\alpha\cos\beta+\cos\alpha\sin\beta)\\
\amp = \cos(\alpha+\beta)+i\sin(\alpha+\beta)\text{.}
\end{align*}
But in exponential notation, this is simply
\begin{equation*}
e^{i\alpha}e^{i\beta} = e^{i(\alpha+\beta)}\text{,}
\end{equation*}
which makes sense, since when you multiply exponentials, you add the exponents.
Generally, problems involving addition and subtraction are best handled in “rectangular” (\(x+iy\)) form, while problems involving multiplication and powers are best handled in polar form.
