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General Definitions of Integral Transforms for Mathematical Physics New Physics: Sae Mulli, Vol. 70, No. 9, September 2020, pp. 759∼765 http://dx.doi.org/10.3938/NPSM.70.759 General Definitions of Integral Transforms for Mathematical Physics Dongseung Kang Department of Mathematics Education, Dankook University, Gyeonggi 16890, Korea Hoewoon Kim Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA Bongwoo Lee∗ Department of Science Education, Dankook University, Gyeonggi 16890, Korea (Received 10 July 2020 : revised 12 August 2020 : accepted 18 August 2020) The Laplace and the Fourier transforms are famous integral transform methods in mathematical physics for solving differential equations. However, most undergraduate textbooks on differential equations only contain a few chapters covering Laplace and Fourier transforms, begin with their definitions as improper integrals on a half line, (0; 1), or on an entire line, (−∞; 1), respectively, proceed with their properties, and then apply them to solve differential equations. Many students just follow these procedures while wondering how those integral transform methods arise naturally for physically reasonable situations. This present article presents new perspectives on derive the general definitions of the Laplace and the Fourier transforms and presents examples in physicsto help students discover the Laplace and the Fourier transforms on various domains, in contrast to only a half line and a whole line in most textbooks. Keywords: Laplace transform, Fourier transform, Integral transforms, Mathematical physics, Differential equations, Undergraduate mathematics I. Introduction physics courses, is in undergraduate physics curriculum to learn and study mathematics also shows that math- Paul A. M. Dirac, an English theoretical physicist, ematics plays an important role in the investigation of communicated his idea of the relation between physics physical sciences. In the course of mathematical physics, and mathematics on presentation of the James Scott integral transforms such as Laplace and Fourier trans- Prize in 1939 saying that there should be two meth- forms are dealt with and provide the students with an ods in the study of natural phenomena making a success opportunity to think about the modern approaches to of research: (1) the method of experiment and observa- solve differential equations as other topics in linear alge- tion and (2) the one of mathematical reasoning [1]. As bra and differential equations. he described the connection between mathematics and In mathematical physics, we use pairs of functions re- physics goes far deeper than the simple perspective of lated by an expression of the following form, “mathematical quality in Nature” and one would agree Z 1 that physics is closely related to mathematics in even g(α) = f(t)K(α; t) dt: −∞ education, not only in research. In particular, the rea- son that mathematical physics, as one of requirements in The function g(α) is called the integral transform of f(t) by the kernel K(α; t) like below: (a) the Laplace integral − ∗E-mail: [email protected] transform with the kernel function K(α; t) = e αt if t > This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. 760 New Physics: Sae Mulli, Vol. 70, No. 9, September 2020 0 and K(α; t) = 0 if t ≤ 0, (b) the Fourier integral From a pedagogical point of view, this approach might transform with the kernel function K(α; t) = e−iαt for make the students confused by the definition because it all t. looks rather strange and demotivating as starting with Integral transforms, especially Laplace transform and an improper integral out of the blue [6, 7]. They are Fourier transform, have many special physical applica- even clueless about what H(s) is, not to mention the tions and interpretations. If an original problem can be applications of it. Students often ask questions like ‘Why solved only with difficulty in the original space, it often do I have to use the Laplace transform to solve an electric happen that integral transform of the problem can be circuit?’ and they might find it difficult to understand solved relatively easily. Then we can get an answer with what they are doing when they use the Laplace transform inverse integral transform. For example, Laplace trans- [8]. From even teacher’s perspectives, not only students’ form can be used to solve ordinary differential equations views, it is one of the most difficult topics for students (ODEs) by reducing a linear differential equation to an to grasp when applying it in applications such as the algebraic equation, which can be easily solved [2]. The electric circuit theory at undergraduate level [8,9]. Fourier transform can also be used to solve partial dif- The integral transform is very important methods be- ferential equations (PDEs). Moreover, because Fourier cause it provides the students with an opportunity to transform decomposes a function (often a function of think about the modern approaches to solve differential time, or a signal) into its constituent frequencies, this equations in terms of operators and functionals. In spite transform enables us to consider physical phenomena of of its importance there seems a lack of researches that time and space in frequency domains and vice versa and could give pedagogical approaches to teach and learn the hence it in turn gives us an interesting insight into the integral transform methods. interpretation of problems in physics. As Joseph Fourier Ngo and Ouzomgi provided a visual way of evaluating introduced the transform in his study of heat transfer, the Laplace transform and its inverse by commutative Fourier transform can be applied to thermal physics, diagrams [10]. However, the article made no considera- acoustics, optics, electrodynamics and quantum mechan- tion of motivation of the transform or any approaches to ics. Of course, integral transforms are not only used for discover and develop it. More recently Randall started the solutions of ODEs but also play an important role with a simple differential equation, solved it by the in- in the analysis of experimental data in physics [3]. The tegrating factor method, and extended the method by use of integral transforms in physics is still one of signifi- applying the concept of the area function for antideriva- cant research areas (e.g. electromagnetic propagation [4] tives to define the Laplace transform [7]. Although he and integral imaging [5].) In particular, the undergradu- tried to convince that his “evolutionary” approach can ate students in scientific fields including physics have to give the understanding of the parameter α in the def- spend considerable time in studying the integral trans- inition of the Laplace transform, there seems no clear forms on the subject of mathematical physics for their explanation of the minus sign just before the parameter majors. α unless we use the same kind of the differential equa- If we describe the process of learning integral trans- tion in his paper. For this matter in 2008 Quinn and forms in mathematical physics by taking an example of Rai presented an approach similar to Randall’s by using Laplace transform, then we are going to: a second order linear differential equation with constant 1. Define the Laplace transform of a piece-wise andof coefficients where they introduced the kernel e−st so that exponential order function f(t) as a function of s, H(s) the involving improper integrals in the process of solving 2. Do some simple examples of Laplace transforms such the equation should be able to converge for the solution as step functions and partial fraction expansions to exist [11]. In 2011 Dwyer offered another way to de- 3. Derive derivatives of Laplace transform fine the Laplace transform by using the inner product of 4. Apply the Laplace transform to simple physics prob- functions in a function space as a vector space with a lems such as simple harmonic oscillator, RLC circuits kernel K(s; t) and deriving a differential equation of the and electromagnetic waves. kernel from the integration by parts [6]. General Definitions of Integral Transforms for Mathematical Physics – Dongseung Kang · Hoewoon Kim 761 @ρ Even though advanced textbooks dealing with the in- (incompressible flow), the equation + r · (ρv) = 0 @t tegral transform such as complex analysis give rigorous reduces to r · v = 0. If we suppose, in addition, that investigations of the formula, they have no motivation the flow is irrotational -that is, r × v = 0 or v = ru for for students. some potential function u, then we obtain a differential The goal of this paper, therefore, is to present an equation (the Laplace equation) alternative approach to the general definitions of the “common” integral transforms. We will present alterna- 52 tive approach to derive the “general” definitions of inte- u = 0 gral transforms on the complex plane with an interesting @2 @2 @2 for a scalar function u and 52 = + + . We example of the Laplace equation in potential theory. @x2 @y2 @z2 shall illustrate how the Laplace transform arises gener- Our paper is organized as follows. The second section ally in this practical application of the Laplace equation of this paper will start with an interesting problem from on a half-plane. As we will realize later we don’t have to physics, solve it, and define the general definitions of take the half space as the domain on which the Laplace Laplace and Fourier transforms on the complex plane equation is considered. However, in order to derive the by analyzing the solution of it. The third section will “general” definition of Laplace transform integrated over explore some examples of Laplace and Fourier transforms the half line we just need to take one of two variable, say on various domains.
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