What is the difference between Laplace and Fourier transforms?

Answer

Dear Rafic,

The Fourier and Laplace transforms are indeed very similar. The Laplace transform is somewhat more general than the Fourier transform. Typically, the Laplace transform is defined on the domain of positive real numbers, although you can also define the bilinear Laplace transform on the full real axis. So that’s not the difference we need to look for.

The Laplace transform is a mapping from real space to part of complex space. It is an image on a right half-plane of the complex plane. The Laplace integral will be convergent (ie exist) for complex values ​​s whose real part is greater than some real number a. We call that number a the convergence abscissa. If the number a is negative, then the Fourier transform also exists. Indeed, because the Fourier transform is actually nothing more than the Laplace transform that you evaluate on the imaginary axis. Consequently, that imaginary axis must be in the region of convergence.

In applied mathematics, when do we use the Laplace transform and when do we use the Fourier transform?

The Laplace transform is used to study dynamical systems such as systems of ordinary differential equations but also some partial differential equations. After all, the Laplace transform ensures that an ordinary differential equation is transformed into an ordinary complex-valued equation. In this way, rational forms or a division of two complex-valued polynomials arise. Using the configuration of the poles and zeros of this rational form, we can study the behavior of the dynamical system. This is a technique that is used in many applications such as electromagnetism, mechanics, econometrics, financial mathematics,…

In principle, one can also do the previous with the Fourier transform, but the poles and zeros almost never lie on that imaginary axis, which makes this less interesting. The Fourier transform is more interesting if one wants to approach the solution of the dynamical system as a function of elementary functions. The elementary functions that the Fourier transform uses are cosines and sines. Under some conditions, these cosines and sines form a (Schauder) basis of the so-called L2 function space, which represents all functions whose square has a finite integral or area. These functions, which are supposedly quadratically integrable, can be approximated as desired by a linear combination of cosines and sines that become increasingly faster. The weights of the linear combination can be calculated using the Fourier transform.

Another transformation that is also of the Fourier/Laplace transform family is the z-transform: since you like a challenge, I challenge you to find out what it is used for and where it differs from the Laplace/Fourier transform. This z-transformation is very important in digital analysis like our digital television, mobile communication and so on…

Indeed, the Fourier and Laplace transforms are not the only transforms available that allow us to approximate or gain insight into functions. This analysis only works well for what we call stationary dynamical systems. Systems whose properties change over time are difficult to describe using Fourier/Laplace transforms. Time-varying systems are typically studied with wavelet transforms that use a basis other than cosines and sines.

Furthermore, the Laplace/Fourier transform works on stationary systems that are supposedly exponentially damping. Some systems are much slower such as polynomial damping systems. For this, “fractional” Laplace/Fourier transformations will be used. It uses a completely different type of derivative called the fractional derivative or Riemann-Liouville differential. Also worth taking a look at this type of (higher) mathematics.

Although those aspects quickly become high level, these methods are very close to scientific research and the state of the art. The courses in higher education where these techniques are discussed are mainly in the Mathematics course, but also in physics and engineering (particularly electrical engineering).

Lots of fun,

Kurt.