Fourier Transform Properties
The Dreaded Laplace Transformation
If you look at the definition of a Laplace transform:
and the Fourier transform:
it may occur to you that the two look remarkably similar. In fact, if you put s = j ω, they are identical, apart from limits. This means that all the Laplace properties suddenly hold for Fourier transforms as well, with little change! This includes the cool way of doing differentiation:
The difference between the two has a large amount to do with the limits of the integrals. Laplace transforms often depend on the initial value of the function; Fourier transforms are independent of the initial value. In fact, the above expression for the Laplace differential only holds when the initial value is zero, while the Fourier one always holds. Also, the transforms are only the same if the function is the same both sides of the y-axis (so the unit step function is different). However, we can still do lots of the same sort of things, including convolution, time-shifting and so on without a lot of difference.
Here, we will investigate some of these properties:
Next, we will take a brief look at using Fourier transforms to solve differential equations.