The complex form of the Fourier Series uses Euler’s identity to express a periodic function as a sum of exponential functions. It simplifies computations and is often used in higher mathematics and physics.
A periodic function \(f(x)\) with period \(2L\) is represented as:
\[ f(x) = \sum_{n=-\infty}^{\infty} c_n \exp\left(\frac{in\pi x}{L}\right) \]
\[ c_n = \frac{1}{2L} \int_{-L}^{L} f(x) \exp\left(\frac{-in\pi x}{L}\right) dx \]