Sketch a graph of \(f(t)=\mathrm{e}^{-|3 t|}\) and find its Fourier transform.

Exponential functions are widely used in science, engineering, and mathematics to represent a variety of phenomena, such as growth, decay, and signal processing. In the case of the function given in our exercise, f(t) = e^{-|3t|}, we deal with both exponential growth and decay, depending on the domain of the independent variable, t.

For t < 0, the function f(t) exhibits exponential growth because the exponent on the natural number e is positive, leading to higher function values as t approaches zero from the negative side. Conversely, for t ≥ 0, we have exponential decay since the exponent is negative, causing the function values to decrease towards zero as t increases.

The graph of an exponential function illustrates these concepts visually, with the curve ascending rapidly on the left for negative t (growth) and descending on the right for positive t (decay), with a peak at t = 0. Understanding the behavior of exponential functions is crucial for analyzing their transformation into the frequency domain using tools like the Fourier transform.

Integral calculus is a fundamental branch of mathematics that deals with the accumulation of quantities and the areas under curves. The process of integration is the reverse of differentiation and is essential to many areas, including probability, area calculations, and, as in this exercise, the analysis of signals with the Fourier transform.

The Fourier transform involves an integral that calculates the frequency spectrum of a time-domain signal. We used integral calculus to compute the integrals of the piecewise components of the given function f(t) = e^{-|3t|}. Specifically, the integration was split into two parts, one for negative t and one for positive t, because of the absolute value in the function's definition.

Integral calculus becomes clearer with practice. The key to solving complex integrals often involves recognizing patterns, like the exponential functions in our case, and applying standard integration techniques such as substitution or, as we did here, relying on well-known formulas for integrating exponentials.

Complex numbers, such as those of the form a + bj where a and b are real numbers, and j is the square root of -1, play a significant role in the field of signal processing and in the computation of the Fourier transform. In the context of the Fourier transform, these numbers allow the description of signals in the frequency domain.

The usage of complex numbers in Fourier transforms is not just a mathematical quirk. It captures both amplitude and phase information of the original time-domain signal. When working with the Fourier transform, we often see terms like e^{-jωt}, where j indicates a 90° phase shift and ω represents angular frequency. The j is crucial because it enables us to incorporate both the sine and cosine components inherent in signals.

In our exercise, we integrated exponential functions containing complex exponents. The results of these integrals were expressed in terms of complex numbers, leading to the final expression for the Fourier transform, which is also in a complex form. This highlights the complex nature of frequencies contributing to the make-up of the original signal.