Why is the Fourier Transform so useful both in theoretical and applied science and engineering?  In short, often it is more convenient to solve a problem in Fourier space than the space of the problem's original formulation.  In this case, one prescription to attack the problem is to convert the representation of the problem to Fourier space, solve the problem in Fourier space (where presumably it is simpler to solve) and then convert back to the original basis. The Fourier Transform may be expressed as: