## Integral Transforms

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 The´orie Analytique de la Chaleur published in 1822. In fact, Laplace’s classic book on La The´orie Analytique des Probabilities included some basic results of the Laplace transform which was 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

with applications. 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. It was the work of Cauchy that contained the exponential form of the Fourier

Integral Theorem as

f(x) = 1

2π

∞∫

∞∫

−∞ eik(x−y)f(y)dydk. (1.1.1)

Cauchy’s work also contained the following formula for functions of the operator D:

φ(D)f(x) = 1

2π

∞∫

∞∫

−∞ φ(ik)eik(x−y)f(y)dydk. (1.1.2)

This essentially led to the modern form of the operational calculus. His famous treatise entitled Memoire sur l’Emploi des Equations Symboliques provided a

deep signiﬁcance Fourier Integral Theorem was recognized by mathematicians and mathematical physicists of the nineteenth and twentieth centuries. Indeed, this theorem is regarded as one of the most fundamental results of modern mathematical analysis and has widespread physical and engineering applications. The generality and importance of the theorem is well expressed by Kelvin and Tait who said: “...Fourier’s Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire, and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.” During the late nineteenth century, it was Oliver Heaviside (1850-1925) who

recognized the power and success of operational calculus and ﬁrst used the operational method as a powerful and eﬀective tool for the solutions of telegraph equation and the second-order hyperbolic partial diﬀerential equations with constant coeﬃcients. In his two papers entitled “On Operational Methods in Physical Mathematics,” Parts I and II, published in The Proceedings of the Royal Society, London, in 1892 and 1893, Heaviside developed operational methods. His 1899 book on Electromagnetic Theory also contained the use and application of the operational methods to the analysis of electrical circuits or networks. Heaviside replaced the diﬀerential operator D≡ ddt by p and treated the latter as an element of the ordinary laws of algebra. The development of his operational methods paid little attention to questions of mathematical rigor. The widespread use of the Heaviside method prior to its vindication by the theory of the Fourier or Laplace transform created a lot of controversy. This was similar to the controversy put forward against the widespread use of the delta function as one of the most useful mathematical devices in Dirac’s logical formulation of quantum mechanics during the 1920s. In fact, P. A. M. Dirac (1902-1984) said: “All electrical engineers are familiar with the idea of a pulse, and the δ-function is just a way of expressing a pulse mathematically.” Dirac’s study of Heaviside’s operator calculus in electromagnetic theory, his training as an electrical engineer, and his deep knowledge of the modern theory of electrical pulses seemed to have a tremendous impact on his ingenious development of modern quantum mechanics. Apparently, the ideas of operational methods originated from the classic

work of Laplace, Fourier, and Cauchy. Inspired by their remarkable works, Heaviside developed his new but less rigorous operational mathematics. In spite of the striking success of Heaviside’s calculus as one of the most useful mathematical methods, contemporary mathematicians hardly recognized Heaviside’s work in his lifetime, primarily due to lack of mathematical rigor. In his lecture on Heaviside and Operational Calculus at the Birth Centenary of Oliver Heaviside, J. L. B. Cooper (1952) revealed some of the controversial issues surrounding Heaviside’s celebrated work, and declared: “As a math-

with a genius for ing convenient methods of calculation. He simpliﬁed Maxwell’s theory enormously; according to Hertz, the four equations known as Maxwell’s were ﬁrst given by Heaviside. He is one of the founders of vector analysis....” Reviewing the history of Heaviside’s calculus, Cooper gave a fairly complete account of early history of the subject along with mathematicians’ varying opinions about Heaviside’s contributions to operational calculus. According to Cooper, a widely publicized story that operational calculus was discovered by Heaviside remained controversial. In spite of the controversies, it is generally believed that Heaviside’s real achievement was to develop operational calculus, which is one of the most useful mathematical devices in applied mathematics, mathematical physics, and engineering science. In this context Lord Rayleigh’s following quotation seems to be most appropriate from a physical point of view: “In the mathematical investigation I have usually employed such methods as present themselves naturally to a physicist. The pure mathematician will complain, and (it must be confessed) sometimes with justice, of deﬁcient rigor. But to this question there are two sides. For, however important it may be to maintain a uniformly high standard in pure mathematics, the physicist may occasionally do well to rest content with arguments which are fairly satisfactory and conclusive from his point of view. To his mind, exercised in a diﬀerent order of ideas, the more severe procedure of the pure mathematician may appear not more but less demonstrative. And further, in many cases of diﬃculty to insist upon highest standard would mean the exclusion of the subject altogether in view of the space that would be required.” With the exception of a group of pure mathematicians, everyone has found

Heaviside’s work a remarkable achievement even though he did not provide a rigorous demonstration of his operational calculus. In defense of Heaviside, Richard P. Feynman’s thought seems to be worth quoting. “However, the emphasis should be somewhat more on how to do the mathematics quickly and easily, and what formulas are true, rather than the mathematicians’ interest in methods of rigorous proof.” The development of operational calculus was somewhat similar to that of calculus of the seventeenth century. Mathematicians who invented the calculus did not provide a rigorous formulation of it. The rigorous formulation came only in the nineteenth century, even though in the transition the non-rigorous demonstration of the calculus that is still admired. It is well known that twentieth-century mathematicians have provided a rigorous foundation of the Heaviside operational calculus. So, by any standard, Heaviside deserves a lot of credit for his remarkable work. The next phase of the development of operational calculus is characterized

by the eﬀort to provide justiﬁcations of the heuristic methods by rigorous proofs. In this phase, T. J. Bromwich (1875-1930) ﬁrst successfully introduced the theory of complex functions to give formal justiﬁcation of Heaviside’s calculus. In addition to his many contributions to this subject, he gave the formal derivation of the Heaviside expansion theorem and the correct interpretation of Heaviside’s operational results. After Bromwich’s work, no-

calculus were by J. R. Carson, B. van der Pol, G. Doetsch, and many others. In concluding our discussion on the historical development of operational

calculus, we should add a note of caution against the controversial evaluation of Heaviside’s work. From an applied mathematical point of view, Heaviside’s operational calculus was an important achievement. In support of his statement, an assessment of Heaviside’s work made by E. T. Whittaker in Heaviside’s obituary is recorded below: “Looking back..., we should place the operational calculus with Poincare´’s discovery of automorphic functions and Ricci’s discovery of the tensor calculus as the three most important mathematical advances of the last quarter of the nineteenth century.” Although Heaviside paid little attention to questions of mathematical rigor, he recognized that operational calculus is one of the most eﬀective and useful mathematical methods in applied mathematical sciences. This has led naturally to rigorous mathematical analysis of integral transforms. Indeed, the Fourier or Laplace transform methods based on the rigorous mathematical foundation are essentially equivalent to the modern operational calculus. There are many other integral transformations including the Mellin trans-

form, the Hankel transform, the Hilbert transform and the Stieltjes transform which are widely used to solve initial and boundary value problems involving ordinary and partial diﬀerential equations and other problems in mathematics, science and engineering. Although, Mellin (1854-1933) presented an elaborate discussion of his transform and its inversion formula, it was G. Bernhard Riemann (1826-1866) who ﬁrst recognized the Mellin transform and its inversion formula in his famous memoir on prime numbers. Hermann Hankel (18391873), a student of G. B. Riemann, introduced the Hankel transform with the Bessel function as its kernel, and this transform can easily be derived from the two-dimensional Fourier transform when circular symmetry is assumed. The Hankel transform arises naturally in solving boundary value problems in cylindrical polar coordinates. Although the Hilbert transform was named after one of the greatest mathe-

maticians of the twentieth century, David Hilbert (1862-1943), this transform and its properties are basically studied by G. H. Hardy (1877-1947) and E. C. Titchmarsh (1899-1963). The Dutch mathematician, T. J. Stieltjes (18561894) introduced the Stieltjes transform in his study of continued fractions. Both the Hilbert and Stieltjes transforms arise in many problems in mathematics, science and engineering. The former is used to solve problems in ﬂuid mechanics, signal processing, and electronics, while the latter arises in solving the integral equations and moment problems.