Engineering Mathematics

18

Fourier Transform

During the study of Fourier series, we confined ourselves to periodic functions. To a periodic function f we assigned Fourier coefficients c_n, n ∈ ℤ and then defined the Fourier series as a trigonometric series with coefficients taken as Fourier coefficients. We then discussed the convergence and some other properties of Fourier series. But we generally encounter non-periodic functions in many applications. Our aim in this chapter is to develop a concept, called Fourier transform, in which to a non-periodic function f, we shall assign for each ω ∈ ℝ a function F defined on ℝ such that F(ω) ∈ ℂ. This function F will be called Fourier transform of the non-periodic function f. The difference, we note, in a Fourier series and ...