Title: 23.2 Representing Periodic Functions by Fourier Series
Course: HELM Workbooks 2023-2024 > Section: HELM 23: Fourier Series

In this Section we show how a periodic function can be expressed as a series of sines and cosines. We begin by obtaining some standard integrals involving sinusoids. We then assume that if \(f(t)\) is a periodic function, of period \(2\pi\), then the Fourier series expansion takes the form: \[ f(t) = \dfrac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos nt + b_n \sin nt)\] Our main purpose here is to show how the constants in this expansion, \(a_n\) (for \(n = 0, 1, 2, 3, \dots\)) and \(b_n\) (for \(n = 1, 2, 3, \dots\)), may be determined for any given function \(f(t)\).