Fourier series: The complex Fourier series
Fourier integrals
Can you work with Fourier series without complex numbers, for Fourier integrals this is practically impossible. Fourier integrals form the analogue of Fourier series in case of non-periodic functions. They can be derived intuitively from Fourier series via a limit transition, but it lacks us here time and space to explain this. Instead, let's simply give the definition. Because the theory is mainly used in signal analysis, we will only consider functions in time \(t\) and will not use the symbol \(x\) as independent variable anymore
Given a 'neat' function \(f(t)\) (we leave aside what 'neat' should mean), the Fourier transform \(\hat{f}(\omega)\) is defined by \[ \hat{f}(\omega) = \int_{-\infty}^{\infty} f(t)\,\e^{-\mathrm{i}\,\omega t}\,\dd t \] This function plays a role similar to that of the Fourier coefficients \(\alpha_n\) of the Fourier series of a periodic function. One calls \(\hat{f}(\omega)\) the spectral density of \(f(t)\).
In a sense, the use of \(\omega\) is unfortunate in this notation, because the variable \(\omega\) plays a different role than the \(\omega\) in the Fourier series. There we had \(\omega = \dfrac{2\pi}{T}\), but here \(\omega\) is a variable that runs through the entire set of real numbers, like the variable \(t\) in the function \(f(t)\). In applications one often speaks about the \(t\)-domain (or the time domain in case of signals) and the \(\omega\)-domain (or the frequency domain). The Fourier transform converts a function \(f(t)\) in the time domain into a function \(\hat{f}(\omega)\) in the frequency domain. The surprising thing is that with this transformation no information is lost, at least when the functions behave decently.
We have already seen how a function in the \(t\)-domain can be transformed to a function in the \(\omega\)-domain. There is also an inverse transform that converts functions in the \(\omega\)-domain back into functions in the \(t\)-domain, and the formula by which this happens is very similar to that of the ordinary Fourier transform: \[ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(\omega) \,\e^{\mathrm{i}\,\omega t}\,\dd\omega\] That is in a sense the analogue of the Fourier series for periodic functions, which indeed rewrites a function as an infinite sum of Fourier coefficients and complex \(e\)-powers. There is of course a lot more to say than is possible here. It suffices to note that the Fourier integral, even more than the Fourier series, is of great importance in applications; it is the main instrument for signal processing.
As an example, we calculate the spectral density \(\hat{s}(\omega)\) of the signal \(s(t)\) given by \[ s(t) = \begin{cases} 1 & \text{if }-\pi\le t\le \pi \\ 0 & \text{otherwise} \end{cases} \] In that case \[\begin{aligned} \hat{s}(\omega) &= \int_{-\infty}^{\infty} s(t) \,\e^{-\mathrm{i}\omega t}\,\dd t\\ \\ &= \int_{-\pi}^{\pi}\e^{-\mathrm{i}\omega t}\, dt \\ \\ &= \left[ \frac{-1}{i\omega} \e^{-\mathrm{i}\omega t} \right]_{t=-\pi}^{\pi} \\ \\ &= \frac{-1}{\mathrm{i}\omega}\left( \e^{-\mathrm{i}\omega \pi} - \e^{\mathrm{i}\omega \pi} \right) \\ \\ &= \frac{2\sin \pi \omega}{\omega}\end{aligned}\]
The inversion formula now gives
\[ s(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{s}(\omega) \, \e^{\mathrm{i}\omega t}\,\dd\omega \]
and in particular is (substitute \(t=0\))
\[ s(0) = 1 = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{p}(\omega) \, \e^{0}\,\dd\omega =
\frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{2\sin \pi \omega}{\omega}\,\dd\omega \] and it follows (set \(x = \pi \omega\) and note that \(\dfrac{\sin x}{x}\) is an even function) the famous result \[ \int_{0}^{\infty} \frac{\sin x}{x}\,\dd x = \frac{\pi}{2}\tiny. \]
The importance of the Fourier transform is only increased in the computer era. The computer made it possible, in principle, to compute numerically the integrals through which the Fourier transform is defined via a so-called Discrete Fourier Transform (DFT). But it became really feasible when the Fast Fourier Transform (FFT) appeared on the scene, an extremely efficient algorithm to switch between the discrete time domain and the discrete frequency domain. All these developments have made it possible to analyse and process continuous and discrete signals in both domains, in almost an unlimited number of applications. Calculus, complex numbers, and complex functions are indispensable tools in all those applications.