Laplace vs Fourier Transforms
Both Laplace transform and Fourier transform are integral transforms, which are most commonly employed as mathematical methods to solve mathematically modelled physical systems. The process is simple. A complex mathematical model is converted in to a simpler, solvable model using an integral transform. Once the simpler model is solved, the inverse integral transform is applied, which would provide the solution to the original model.
For example, since most of the physical systems result in differential equations, they can be converted into algebraic equations or to lower degree easily solvable differential equations using an integral transform. Then solving the problem will become easier.
What is the Laplace transform?
Given a function f (t) of a real variable t, its Laplace transform is defined by the integral [latex] F(s) = \\int_{0}^{ \\infty} e^{-st}f(t)dt [/latex] (whenever it exists), which is a function of a complex variable s. It is usually denoted by L {f (t)}. The inverse Laplace transform of a function F(s) is taken to be the function f (t) in such a way that L {f (t)} = F(s), and in the usual mathematical notation we write, L -1{F(s)} = f (t). The inverse transform can be made unique if null functions are not allowed. One can identify these two as linear operators defined in the function space, and it is also easy to see that, L -1{ L {f (t)}} = f (t), if null functions are not allowed.
The following table lists the Laplace transforms of some of most common functions.
What is the Fourier transform?
Given a function f (t) of a real variable t, its Laplace transform is defined by the integral [latex] F( \\alpha )= \\frac{1}{\\sqrt{2 \\pi}} \\int_{- \\infty}^{\\infty} e^{i \\alpha t}f(t)dt [/latex] (whenever it exists), and is usually denoted by F { f (t)}. The inverse transform F -1{F(α)} is given by the integral [latex] f(t)=\\frac{1}{\\sqrt{2 \\pi}}\\int_{- \\infty}^{\\infty} e^{-i \\alpha t}F(\\alpha )d\\alpha [/latex]. Fourier transform is also linear, and can be thought of as an operator defined in the function space.
Using the Fourier transform, the original function can be written as follows provided that the function has only finite number of discontinuities and is absolutely integrable.
What is the difference between the Laplace and the Fourier Transforms?
- Fourier transform of a function f (t) is defined as [latex] F( \\alpha )= \\frac{1}{\\sqrt{2 \\pi}} \\int_{- \\infty}^{\\infty} e^{i \\alpha t}f(t)dt [/latex], whereas the laplace transform of it is defined to be [latex] F(s) = \\int_{0}^{ \\infty} e^{-st}f(t)dt [/latex].
- Fourier transform is defined only for functions defined for all the real numbers, whereas Laplace transform does not require the function to be defined on set the negative real numbers.
- Fourier transform is a special case of the Laplace transform. It can be seen that both coincide for non-negative real numbers. (i.e. take s in the Laplace to be iα + β where α and β are real such that e β= 1/√(2ᴫ))
- Every function that has a Fourier transform will have a Laplace transform but not vice-versa.