Fourier series are a mathematical tool that expresses periodic functions as an infinite sum of sine and cosine functions.
A periodic function \(f(x)\) with period \(T\) can be represented by a Fourier series in its general form as:
\[ f(x) = \sum_{k=0}^{\infty} \left( a_k \cos\left(\frac{2\pi k x}{T}\right) + b_k \sin\left(\frac{2\pi k x}{T}\right) \right), \]
where the coefficients \(a_k\) and \(b_k\) are calculated as:
- \(a_0 = \frac{1}{T} \int_{0}^{T} f(x)\,dx\) (average component).
- \(a_k = \frac{2}{T} \int_{0}^{T} f(x) \cos\left(\frac{2\pi k x}{T}\right)\,dx\), for \(k \geq 1\).
- \(b_k = \frac{2}{T} \int_{0}^{T} f(x) \sin\left(\frac{2\pi k x}{T}\right)\,dx\), for \(k \geq 1\).
The sum starts with \(k=0\), where the term \(a_0\) represents the average component (or constant term) of the function, while the terms with \(k \geq 1\) correspond to the harmonic frequencies.
Fourier series allow decomposing a function into its harmonic components, facilitating analysis in the frequency domain. This tool is essential in fields like signal processing, acoustics, solving differential equations, and more.