Fourier Series

x 1 and fails to satisfy the Dirichlet Conditions.Example 3Find the whole-range Fourier series that represents the function fx defined in −, byfx 2 − x 0 x − 2 0 ≤ x .By evaluating the series at x 0,, deduce that∑n11 2n 2 6 , ∑n1−1n−1n 2 212 .We wish to represent the function fx asfx ~ Sx 1 2 a 0 ∑n1a n cosnx b n sinnxand the formulae for the Fourier coefficients are already known. The first two steps are todraw the function to ascertain its properties and to check that it satisfies the Dirichletconditions.Figure 1.3.2. Schematic of given function.The function is shown in figure 1.3.2 above and it is straightforward to verify that itsatisfies the Dirichlet conditions. Furthermore, from the figure, it is clear that the functionis neither even nor odd and so we expect both cosine and sine terms to be present.Determining the coefficients from the given formulae.20