Title: 23.6 The Complex form
Course: HELM Workbooks 2023-2024 > Section: HELM 23: Fourier Series

In this Section we show how a Fourier series can be expressed more concisely if we introduce the complex number \(i\) where \(i^2 = -1\). By utilising the Euler relation: \[ e^{i\theta} \equiv \cos \theta + i \sin \theta \]

we can replace the trigonometric functions by complex exponential functions. By also combining the Fourier coefficients \(a_n\) and \(b_n\) into a complex coefficient \(c_n\) through \[ c_n = \dfrac{1}{2} \left( a_n - i b_n \right) \]

we find that, for a given periodic signal, both sets of constants can be found in one operation.

We also obtain Parseval's theorem which has important applications in electrical engineering.

The complex formulation of a Fourier series is an important precursor of the Fourier transform which attempts to Fourier analyse non-periodic functions.