8.1: Complex Representations of Waves

We have seen that we can determine the frequency content of a function \(f(t)\) defined on an interval \([0, T]\) by looking for the Fourier coefficients in the Fourier series expansion \[f(t)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos \frac{2 \pi n t}{T}+b_{n} \sin \frac{2 \pi n t}{T} .\nonumber \] The coefficients take forms like \[a_{n}=\frac{2}{T} \int_{0}^{T} f(t) \cos \frac{2 \pi n t}{T} d t .\nonumber \]

However, trigonometric functions can be written in a complex exponential form. Using Euler’s formula, which was obtained using the Maclaurin expansion of \(e^{x}\) in Example 11.7.8, \[e^{i \theta}=\cos \theta+i \sin \theta \text {, }\nonumber \] the complex conjugate is found by replacing \(i\) with \(-i\) to obtain \[e^{-i \theta}=\cos \theta-i \sin \theta \text {. }\nonumber \] Adding these expressions, we have \[2 \cos \theta=e^{i \theta}+e^{-i \theta} \text {. }\nonumber \] Subtracting the exponentials leads to an expression for the sine function. Thus, we have the important result that sines and cosines can be written as complex exponentials: \[\begin{align} &\cos \theta=\frac{e^{i \theta}+e^{-i \theta}}{2}\nonumber \\ &\sin \theta=\frac{e^{i \theta}-e^{-i \theta}}{2 i}\label{eq:1} \end{align}\]

So, we can write \[\cos \frac{2 \pi n t}{T}=\frac{1}{2}\left(e^{\frac{2 \pi i n t}{T}}+e^{-\frac{2 \pi i n t}{T}}\right) .\nonumber \] Later we will see that we can use this information to rewrite the series as a sum over complex exponentials in the form \[f(t)=\sum_{n=-\infty}^{\infty} c_{n} e^{\frac{2 \pi i n t}{T}},\nonumber \] where the Fourier coefficients now take the form \[c_{n}=\int_{0}^{T} f(t) e^{-\frac{2 \pi i n t}{T}} d t .\nonumber \] In fact, when one considers the representation of analogue signals defined over an infinite interval and containing a continuum of frequencies, we will see that Fourier series sums become integrals of complex functions and so do the Fourier coefficients. Thus, we will naturally find ourselves needing to work with functions of complex variables and perform complex integrals.

We can also develop a complex representation for waves. Recall from the discussion in Section 3.6 on finite length strings that a solution to the wave equation was given by \[u(x, t)=\frac{1}{2}\left[\sum_{n=1}^{\infty} A_{n} \sin k_{n}(x+c t)+\sum_{n=1}^{\infty} A_{n} \sin k_{n}(x-c t)\right] .\label{eq:2} \] We can replace the sines with their complex forms as \[\begin{align} u(x, t)=& \frac{1}{4 i}\left[\sum_{n=1}^{\infty} A_{n}\left(e^{i k_{n}(x+c t)}-e^{-i k_{n}(x+c t)}\right)\right.\nonumber \\ &\left.+\sum_{n=1}^{\infty} A_{n}\left(e^{i k_{n}(x-c t)}-e^{-i k_{n}(x-c t)}\right)\right] .\label{eq:3} \end{align}\]

Defining \(k_{-n}=-k_{n}, n=1,2, \ldots\), we can rewrite this solution in the form \[u(x, t)=\sum_{n=-\infty}^{\infty}\left[c_{n} e^{i k_{n}(x+c t)}+d_{n} e^{i k_{n}(x-c t)}\right] .\label{eq:4}\]

Such representations are also possible for waves propagating over the entire real line. In such cases we are not restricted to discrete frequencies and wave numbers. The sum of the harmonics will then be a sum over a continuous range, which means that the sums become integrals. So, we are lead to the complex representation \[u(x, t)=\int_{-\infty}^{\infty}\left[c(k) e^{i k(x+c t)}+d(k) e^{i k(x-c t)}\right] d k .\label{eq:5}\]

The forms \(e^{i k(x+c t)}\) and \(e^{i k(x-c t)}\) are complex representations of what are called plane waves in one dimension. The integral represents a general wave form consisting of a sum over plane waves. The Fourier coefficients in the representation can be complex valued functions and the evaluation of the integral may be done using methods from complex analysis. We would like to be able to compute such integrals.

With the above ideas in mind, we will now take a tour of complex analysis. We will first review some facts about complex numbers and then introduce complex functions. This will lead us to the calculus of functions of a complex variable, including the differentiation and integration complex functions. This will set up the methods needed to explore Fourier transforms in the next chapter.