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Section 3.2 Complex exponentials and trig functions

Subsection 3.2.1 Complex numbers and arithmetic

I am a complex analyst by trade, so I will assert that the complex numbers are really the only way to go if you want to do mathematical analysis. The use of complex numbers and functions to represent real systems is common convention for several reasons. First, the math is WAY easier in the complex setting. Basically, without delving into complex analysis, we're going to be cheating by thinking of functions as complex-valued with real inputs.

First and foremost, I am a mathematician, not an engineer, so I will use the symbol \(i\) to represent the terribly named imaginary unit

\begin{equation*} i = \sqrt{-1}. \end{equation*}

Feel free to use \(j\) if it makes your brain happy.

A complex number is a quantity

\begin{equation*} z = x + iy \end{equation*}

where \(x, y\) are real numbers. \(x\) is called the real part of \(z\) and is written \(x = \RE z\text{.}\) Likewise, \(y\) is called the imaginary part of \(z\) (yes, the imaginary part is real) and is written \(y = \IM z\text{.}\) Notice that this means that we can think of the real numbers as living inside the complex numbers (it's the set with \(y = 0\)).

You should think of the complex numbers as essentially having the same structural features as the real numbers. The operations are defined on the real and imaginary parts, with the additional feature that \(i^2 = -1\text{.}\) Say that \(z = x + iy\) and \(w = a + ib\text{.}\)

  • Addition:
    \begin{equation*} z + w = (x + a) + i(y + b) \end{equation*}
  • Multiplication
    \begin{equation*} zw = (a + ib)(x + iy) = (ax - by) + i(bx + ay) \end{equation*}
  • Division:
    \begin{equation*} \frac{z}{w} = \frac{x + iy}{a + ib} = \frac{(x + iy)(a - ib)}{(a + ib)(a - ib)} = \frac{(ax + by)}{a^2 + b^2} + i \frac{ay - bx}{a^2 + b^2} \end{equation*}

If \(z = x + iy\) then the complex conjugate of \(z\) is

\begin{equation*} \cc{z} = x - iy. \end{equation*}

Complex conjugates follow the following “distribution” like rules:

\begin{equation*} \cc{z + w} = \cc{z} + \cc{w}, \,\, \cc{zw} = \cc{z} \cc{w}, \,\, \cc{\cc{z}} =z. \end{equation*}

With complex conjugates, it is very easy to write formulas for the real and imaginary part of a complex number.

\begin{equation*} \RE z = \frac{z + \cc{z}}{2}, \,\,\, \IM z = \frac{z - \cc{z}}{2i} \end{equation*}

Complex conjugates are very important in calculus and matrix settings, so we have an alternative notation that we apply to more complicated objects: \(\cc{z} = z\ad\text{.}\) Because integrals are limits of finite sums, integration respects complex conjugation and we write

\begin{equation*} \left(\int f(t)g(t) \, dt \right)\ad = \int f(t)\ad g(t)\ad \, dt. \end{equation*}

A natural way to think of a complex number is as a vector, with the real part representing the “\(x\)-coordinate” and the imaginary part the “\(y\)”. Because of the way that complex conjugates are defined, we can compute the norm or magnitude of a complex number by

\begin{equation*} \abs{z} = \sqrt{x^2 + y^2} = \sqrt{z \cc{z}}. \end{equation*}

In practice, it is useful to write the above expression as \(\abs{z}^2 = z \cc{z}\text{.}\)

Another useful related property is

\begin{equation*} \abs{z + w}^2 = \abs{z}^2 + 2\RE z\cc{w} + \abs{w}^2. \end{equation*}

Finally, just as a vector in \(\R^2\) can be thought of in both Cartesian and polar forms, so too can a complex number. Given \(z = x + iy\text{,}\) the length of the vector can be written \(r = \abs{z} = \sqrt{x^2 + y^2}\) and the angle from the positive real axis satisifies \(\tan \theta = \frac{y}{x}\text{.}\) The angle \(\theta\) is called the argument of the complex number and is sometimes written \(\arg z\text{.}\) First note that this means that the complex numbers of radius 1 coincide precisely with the unit circle. Since \(x = r \cos \theta\) and \(y = r \sin \theta\text{,}\) we can write

\begin{equation*} z = r\cos\theta + i r \sin \theta, \end{equation*}

though we're quickly going to get a more useful form. The thing to note is that complex numbers naturally encode and support the trig functions that represent rotations. As complicated as complex numbers might seem, this is the major theme of our use of them: complex numbers are rotations of real numbers.

Subsection 3.2.2 Complex exponentials and Euler's formula

The most important complex function is the complex exponential \(f(z) = e^z\text{.}\) Given a Taylor series with a positive interval of convergence, we can extend the Taylor series formula to complex numbers simply by plugging in complex numbers. The set of convergence will be a disk with diameter equivalent to the real interval of convergence. Hence, we make the definition

\begin{equation*} e^z = \sum_{n=0}^\infty \frac{z^n}{n!}, \end{equation*}

which converges for all \(z \in \C\text{.}\) It is an easy exercise (do it!) to show that

\begin{equation*} \cc{e^z} = e^{\cc{z}}. \end{equation*}

As is the case in one variable, \(e^z\) is the unique function that is its own derivative. We will take on authority that the complex exponential satisfies the usual algebraic properties (as defining the complex logarithm is more appropriately done in a complex analysis course):

\begin{align*} e^{z + w} \amp= e^z + e^w\\ {e^z}^w \amp e^{zw} \end{align*}

Using the properties of exponential functions, we can write

\begin{equation*} e^z = e^{x + iy} = e^x e^{iy}. \end{equation*}

This doesn't seem like we've gained much, but something magical happens with \(e^{iy}\text{.}\) When the power of a complex exponential is purely imaginary, it corresponds to a point on the unit circle! That is,

\begin{equation} e^{iy} = \cos y + i \sin y,\label{eq-cis}\tag{3.2.1} \end{equation}

which is an almost unimaginably useful formula called Euler's formula. You might suspect such a relationship should hold given that the Taylor series for sine and cosine almost look like they add to \(e^x\) in the real case, but the alternating signs are wrong. In the complex case, this relation holds.

What this means is that complex multiplication and division (and hence exponentiation) can be rewritten in form that is tractable. If \(z_1 = r_1 e^{i\theta_1}, z_2 = r_2 e^{i\theta_2}\text{,}\) then

\begin{equation*} z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}, \end{equation*}

and

\begin{equation*} \frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i \theta_1 - \theta_2}. \end{equation*}

It's easy to miss how weird and useful this idea is. Essentially, we've come up with a way to multiply vectors and produce a vector.

Since (3.2.1) expresses \(e^{i\theta}\) in terms of sine and cosine, we can reverse this relation to define sine and cosine in terms of complex exponentials.

\begin{equation} \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}, \,\,\, \sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}.\label{eq-sinu-exp}\tag{3.2.2} \end{equation}

One of the themes of this course is that we can decompose functions into pieces that have useful properties. One way that you might not have seen before is to pull a function apart into symmetric pieces. A function is called even if \(f(x) = f(-x)\) (corresponding to symmetry about the \(y\)-axis. A function is called odd if \(f(x) = -f(-x)\) (corresponding to symmetry about the origin). We can build even and odd functions out of any function in the following way:

\begin{equation*} f_{even}(x) = \frac{f(x) + f(-x)}{2}, \,\,\, f_{odd}(x) = \frac{f(x) - f(-x)}{2}. \end{equation*}
\(f_{even}\)\(f_{odd}\)

\(f_{even}\) and \(f_{odd}\) decompose any function into symmertic pieces, as

\begin{equation*} f_{even}(x) + f_{odd}(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2} = f(x). \end{equation*}

It turns out that the even and odd pieces of the complex exponential are the trig functions.

\begin{equation*} \frac{e^{i\theta} + e^{-i\theta}}{2} = \cos \theta, \end{equation*}

and

\begin{equation*} \frac{e^{i\theta} - e^{-i\theta}}{2} = i \sin \theta. \end{equation*}

Subsection 3.2.3 What complex exponentials mean and how to use them

Typically, we're going to be working with a real parameter \(t\text{.}\) So define the function

\begin{equation*} f(t) = e^{it} = \cos t + i \sin t. \end{equation*}

As \(t\) moves along in time, what is \(f\) doing? If we think about the complex numbers as vectors (that is, \(f(t) = (\RE f(t), \IM f(t))\text{,}\) we can think of \(f\) as a parametric function where \(x(t) = \cos t\) and \(y(t) = \sin t\text{.}\) Then as \(t\) increases, the graph of \(f\) traces out the unit circle, rotating in the counterclockwise direction. Likewise, negative complex exponentials trace out the unit circle but in the clockwise direction. One complete traversal of the circle takes place as \(t\) goes from \(0\) to \(2 \pi\text{.}\)

It is typical practice in applications to work with integer values for frequency, and so in engineering (and general Fourier analysis), the convention is to use as the basic exponential relation

\begin{equation*} e^{2\pi i t} = \cos 2\pi t + i \sin 2\pi t, \end{equation*}

as this gives a function with frequency \(1\) (that is, we complete one loop around the unit circle for \(t \in [0,1]\)). In other words, \(e^{2\pi i t}\) has frequency \(1 \hz\text{.}\) More generally,

\begin{equation*} e^{2\pi n i t} = \cos{2 \pi n t} + i \sin 2\pi n t \end{equation*}

has frequency \(n \hz\text{.}\)

Notice that \(e^{-2\pi n i t}\) also has frequency \(n \hz\text{,}\) but travels in the opposite direction. This will be an important theme in Fourier analysis - oscillations of a given frequency have two components, propagating in opposite directions.

Complex exponentials are vastly easier to work with than the sinusoids. Recall that the generic sine function is

\begin{equation*} f(t) = A \sin(2 \pi \nu t + \phi) \end{equation*}

with frequency \(\nu\text{,}\) amplitude \(A\text{,}\) and phase \(\phi\text{.}\) The complex exponential with the same information is

\begin{equation*} g(t) = A e^{i(2\pi \nu t + \phi)} \end{equation*}

which can be split into

\begin{equation*} g(t) = Ae^{i\phi} e^{2 \pi \nu u t}. \end{equation*}

Notice that

\begin{equation*} \abs{Ae^{i\phi} e^{2 \pi \nu u t}} = \abs{A} = A \end{equation*}

and that \(e^{i\phi}\) describes a starting point on the unit circle.

Finally, we'll come to a fact that you may have seen in previous courses, though usually without explanation.

This is not at all obvious from a superimposed graph.

But if we plot the sum, we get:

Here the computation is massively simplified by the use of complex exponentials. Consider

\begin{equation*} A_1 e^{i(2 \pi \nu + \phi_1)} + A_2 e^{i(2 \pi \nu + \phi_2)}. \end{equation*}

Using basic exponential algebra, we can compute

\begin{align*} \amp A_1 e^{i(2 \pi \nu + \phi_1)} + A_2 e^{i(2 \pi \nu + \phi_2)}\\ \amp = e^{2 \pi i \nu t}\left(A_1 e^{i\phi_1} + A_2 e^{i\phi_2}\right). \end{align*}

That is, the sum of two exponentials with the same frequency is another exponential of the same frequency (but with a new phase and amplitude).