The thorough study of nature is the most fertile ground for mathematical discoveries.
- Joseph Fourier

If you wish to foresee the future of mathematics our proper course is to study the history and present condition of the science.
- Henri Poincar´e

The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics, it is mathematics which builds the linking bridges and gives the ever more reliable forms. From this it has come about that our entire contemporary culture, in as much as it is based the intellectual penetration and the exploitation of nature, has its foundations in mathematics.
- David Hilbert

Integral transformations have been successfully used for almost two centuries in solving many problems in applied mathematics, mathematical physics, and engineering science. Historically, the origin of the integral transforms including the Laplace and Fourier transforms can be traced back to celebrated work of P.S.Laplace (1749-1827) on probability theory in the 1780s and to monumental treatise of Joseph Fourier (1768-1830) on La Th´eorie Analytique de la Chaleur published in 1822. In fact, Laplace’s classic book on La Th´eorie Analytique des Probabilities includes some basic results of the Laplace transform which is one of the oldest and most commonly used integral transforms available in the mathematical literature. This has effectively been used in finding the solution of linear differential equations and integral equations. On the other hand, Fourier’s treatise provided the modern mathematical theory of heat conduction, Fourier series, and Fourier integrals with applications. In his treatise, Fourier stated a remarkable result that is universally known as the Fourier Integral Theorem. He gave a series of examples before stating that an arbitrary function defined on a finite interval can be expanded in terms of trigonometric series which is now universally known as the Fourier series. In an attempt to extend his new ideas to functions defined on an infinite interval, Fourier discovered an integral transform and its inversion formula which are now well known as the Fourier transform and the inverse Fourier transform. However, this celebrated idea of Fourier was known to Laplace and A. L. Cauchy (1789-1857) as some of their earlier work involved this transformation. On the other hand, S. D. Poisson (1781-1840) also independently used the method of transform in his research on the propagation of water waves.

However, it was G.W.Leibniz (1646-1716) who first introduced the idea of a symbolic method in calculus. Subsequently, both J.L.Lagrange (1736-1813) and Laplace made considerable contributions to symbolic methods which became known as operational calculus. Although both the Laplace and the Fourier transforms have been discovered in the nineteenth century, it was the British electrical engineer Oliver Heaviside (1850-1925) who made the Laplace transform very popular by using it to solve ordinary differential equations of electrical circuits and systems, and then to develop modern operational calculus. It may be relevant to point out that the Laplace transform is essentially a special case of the Fourier transform for a class of functions defined on the positive real axis, but it is more simple than the Fourier transform for the following reasons. First, the question of convergence of the Laplace transform is much less delicate because of its exponentially decaying kernel exp (-st), where Re s>0 and t>0. Second, the Laplace transform is an analytic function of the complex variable and its properties can easily be studied with the knowledge of the theory of complex variable. Third, the Fourier integral formula provided the definitions of the Laplace transform and the inverse Laplace transform in terms of a complex contour integral that can be evaluated with the help the Cauchy residue theory and deformation of contour in the complex plane.

If you want to know more, you can read this nice book…
Debnath, L. and Bhatta, D., 2007, “Integral Transforms and Their Applications 2ed“, Taylor & Francis Group, LLC.

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