Question

Find the Fourier series of the periodic function defined by: 

f(x) = 0, if − π ≤ x ≤ 0 

f(x) = π, if 0 ≤ x ≤ π

Answer 1

The given function is of the square form. As f (x) is defined in the interval (−π, π), the Fourier expansion is given by

The graph of f (x) is shown in Fig. 1.8. It consists of the x-axis from −π to 0 and of the line AB from 0 to π. A simple discontinuity occurs at x = 0 at which point the series reduces to π/2. 

Now, π/2 = 1/2[ f (0−) + f (0+)]

which is consistent with Dirichlet’s theorem. Similar behavior is exhibited at x = π, ±2π... Figure 1.8 shows first four partial sums with equations 

y = π/2 

y = π/2 + 2 sin x 

y = π/2 + 2(sin x + (1/3) sin 3x) 

y = π/2 + 2(sin x + (1/3) sin 3x + (1/5) sin 5x)