Fourier analysis

Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric functions. What if the interval is infinite? That's the subject of this chapter. Instead of a sum over frequencies, you will have an integral. 15.1 Fourier Transform For the finite interval you have to specify the boundary conditions in order to determine the particular basis that you're going to use. On the infinite interval you don't have this large set of choices. After all, if the boundary is infinitely far away, how can it affect what you're doing over a finite distance? But see section 15.6. In section 5.3 you have several boundary condition listed that you can use on the differential equation u = λu and that will lead to orthogonal functions on your interval. For the purposes here the easiest approach is to assume periodic boundary conditions on the finite interval and then to take the limit as the length of the interval approaches infinity. On −L < x < +L, the conditions on the solutions of u = λu are then u(−L) = u(+L) and u (−L) = u (+L). The solution to this is most conveniently expressed as a complex exponential, Eq. (5.19)