The Fourier Series allows a periodic function to be expressed as a sum of sine and cosine functions. This representation is especially helpful in analyzing signals and solving differential equations. The real form uses trigonometric terms and is suitable when dealing with real-valued functions.
A function \( f(x) \) defined on an interval \([-L, L]\) with period \(2L\) can be written as:
\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos \frac{n\pi x}{L} + b_n \sin \frac{n\pi x}{L} \right) \]
\[ a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \frac{n\pi x}{L} \, dx \]
\[ b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \frac{n\pi x}{L} \, dx \]