To solve a complex equation involving Fourier transforms, let's break down the process step by step. We'll start with a basic understanding of Fourier transforms and then work through a specific example. Understanding Fourier Transforms Fourier transforms are mathematical tools used to analyze functions or signals in terms of frequencies. They decompose a function into its constituent frequencies, much like a musical chord can be expressed as the frequencies of its individual notes. Key Concept: - Fourier Transform (FT): Converts a time-domain signal into its frequency-domain representation. - Inverse Fourier Transform: Converts a frequency-domain signal back into the time domain. Example Problem Let's consider a simple example: \( f(t) = e^{-at} \), where \( a > 0 \) and \( t \) is time. Objective: Find the Fourier Transform of \( f(t) \). Step-by-Step Solution 1. Define the Fourier Transform: The Fourier Transform of a function \( f(t) \) is given by \( F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt \), where \( \omega \) is angular frequency, and \( i \) is the imaginary unit. 2. Apply the Fourier Transform to \( f(t) \): - Substitute \( f(t) \) into the Fourier Transform equation: \( F(\omega) = \int_{-\infty}^{\infty} e^{-at} e^{-i\omega t} dt \). 3. Combine the Exponents: - Simplify the expression: \( F(\omega) = \int_{-\infty}^{\infty} e^{-(a + i\omega) t} dt \). 4. Solve the Integral: - This is an exponential integral, which can be solved as: \( F(\omega) = \left[ \frac{e^{-(a + i\omega) t}}{-(a + i\omega)} \right]_{-\infty}^{\infty} \). 5. Evaluate the Limits: - Since \( a > 0 \), the exponential term goes to zero as \( t \) goes to infinity. Therefore, the integral simplifies to: \( F(\omega) = \frac{1}{a + i\omega} \). 6. Interpret the Result: - The result, \( F(\omega) = \frac{1}{a + i\omega} \), represents the frequency-domain representation of \( f(t) \). It shows how the exponential decay in the time domain translates into a frequency spectrum. Conclusion By following these steps, we've transformed a time-domain signal into its frequency-domain representation using the Fourier Transform. This process is fundamental in various fields, including signal processing, engineering, and physics, to analyze the frequency components of signals and functions.