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,∞), or on an entire line, (−∞, ∞), 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 physics to 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
Paul A. M. Dirac, an English theoretical physicist, communicated his idea of the relation between physics and mathematics on presentation of the James Scott Prize in 1939 saying that there should be two meth- ods in the study of natural phenomena making a success of research: (1) the method of experiment and observa- tion and (2) the one of mathematical reasoning [1]. As he described the connection between mathematics and physics goes far deeper than the simple perspective of
“mathematical quality in Nature” and one would agree that physics is closely related to mathematics in even education, not only in research. In particular, the rea- son that mathematical physics, as one of requirements in
∗E-mail: [email protected]
physics courses, is in undergraduate physics curriculum to learn and study mathematics also shows that math- ematics plays an important role in the investigation of physical sciences. In the course of mathematical physics, integral transforms such as Laplace and Fourier trans- forms are dealt with and provide the students with an opportunity to think about the modern approaches to solve differential equations as other topics in linear alge- bra and differential equations.
In mathematical physics, we use pairs of functions re- lated by an expression of the following form,
g(α) =
∫ ∞
−∞
f (t)K(α, t) dt.
The function g(α) is called the integral transform of f (t) by the kernel K(α, t) like below: (a) the Laplace integral transform with the kernel function K(α, t) = e−αtif t >
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0 and K(α, t) = 0 if t ≤ 0, (b) the Fourier integral transform with the kernel function K(α, t) = e−iαt for all t.
Integral transforms, especially Laplace transform and Fourier transform, have many special physical applica- tions and interpretations. If an original problem can be solved only with difficulty in the original space, it often happen that integral transform of the problem can be solved relatively easily. Then we can get an answer with inverse integral transform. For example, Laplace trans- form can be used to solve ordinary differential equations (ODEs) by reducing a linear differential equation to an algebraic equation, which can be easily solved [2]. The Fourier transform can also be used to solve partial dif- ferential equations (PDEs). Moreover, because Fourier transform decomposes a function (often a function of time, or a signal) into its constituent frequencies, this transform enables us to consider physical phenomena of time and space in frequency domains and vice versa and hence it in turn gives us an interesting insight into the interpretation of problems in physics. As Joseph Fourier introduced the transform in his study of heat transfer, Fourier transform can be applied to thermal physics, acoustics, optics, electrodynamics and quantum mechan- ics. Of course, integral transforms are not only used for the solutions of ODEs but also play an important role in the analysis of experimental data in physics [3]. The use of integral transforms in physics is still one of signifi- cant research areas (e.g. electromagnetic propagation [4]
and integral imaging [5].) In particular, the undergradu- ate students in scientific fields including physics have to spend considerable time in studying the integral trans- forms on the subject of mathematical physics for their majors.
If we describe the process of learning integral trans- forms in mathematical physics by taking an example of Laplace transform, then we are going to:
1. Define the Laplace transform of a piece-wise and of exponential order function f (t) as a function of s, H(s) 2. Do some simple examples of Laplace transforms such as step functions and partial fraction expansions 3. Derive derivatives of Laplace transform
4. Apply the Laplace transform to simple physics prob- lems such as simple harmonic oscillator, RLC circuits and electromagnetic waves.
From a pedagogical point of view, this approach might make the students confused by the definition because it looks rather strange and demotivating as starting with an improper integral out of the blue [6,7]. They are even clueless about what H(s) is, not to mention the applications of it. Students often ask questions like ‘Why do I have to use the Laplace transform to solve an electric circuit? ’ and they might find it difficult to understand what they are doing when they use the Laplace transform [8]. From even teacher’s perspectives, not only students’
views, it is one of the most difficult topics for students to grasp when applying it in applications such as the electric circuit theory at undergraduate level [8,9].
The integral transform is very important methods be- cause it provides the students with an opportunity to think about the modern approaches to solve differential equations in terms of operators and functionals. In spite of its importance there seems a lack of researches that could give pedagogical approaches to teach and learn the integral transform methods.
Ngo and Ouzomgi provided a visual way of evaluating the Laplace transform and its inverse by commutative diagrams [10]. However, the article made no considera- tion of motivation of the transform or any approaches to discover and develop it. More recently Randall started with a simple differential equation, solved it by the in- tegrating factor method, and extended the method by applying the concept of the area function for antideriva- tives to define the Laplace transform [7]. Although he tried to convince that his “evolutionary” approach can give the understanding of the parameter α in the def- inition of the Laplace transform, there seems no clear explanation of the minus sign just before the parameter α unless we use the same kind of the differential equa- tion in his paper. For this matter in 2008 Quinn and Rai presented an approach similar to Randall’s by using a second order linear differential equation with constant coefficients where they introduced the kernel e−stso that the involving improper integrals in the process of solving the equation should be able to converge for the solution to exist [11]. In 2011 Dwyer offered another way to de- fine the Laplace transform by using the inner product of functions in a function space as a vector space with a kernel K(s, t) and deriving a differential equation of the kernel from the integration by parts [6].
Even though advanced textbooks dealing with the in- tegral transform such as complex analysis give rigorous investigations of the formula, they have no motivation for students.
The goal of this paper, therefore, is to present an alternative approach to the general definitions of the
“common” integral transforms. We will present alterna- tive approach to derive the “general” definitions of inte- gral transforms on the complex plane with an interesting example of the Laplace equation in potential theory.
Our paper is organized as follows. The second section of this paper will start with an interesting problem from physics, solve it, and define the general definitions of Laplace and Fourier transforms on the complex plane by analyzing the solution of it. The third section will explore some examples of Laplace and Fourier transforms on various domains. The last section as a conclusion makes some discussions and remarks about the integral transform methods for differential equations.
II. General Definitions of Laplace and Fourier Transforms
We begin this section for the general defini- tion of Laplace transform by considering a region D ⊂ R3 filled with a fluid having variable den- sity ρ(x, y, z, t) (g/cm3) and the flow of the fluid is defined by the vector velocity field v(x, y, z, t) =
⟨v1(x, y, z, t), v2(x, y, z, t), v3(x, y, z, t)⟩ (cm/s). By the principle of conservation of mass the rate of increase of mass in a fixed subregion W of D (W does not change with time) equals the rate at which mass is crossing the boundary of W , ∂W in the inward direction; i.e.,
d dt
∫
W
ρ dV =−
∫
∂W
ρv· n dA (1) where n is the outward normal vector to the boundary of W , ∂W , assuming there is no distribution of mass source.
This is the integral form of the law of conservation of mass. Using the divergence (or Gauss) theorem we show that the integral form (1) above is equivalent to
∫
W
{∂ρ
∂t +∇ · (ρv) }
dV = 0 (2)
and hence ∂ρ
∂t +∇ · (ρv) = 0 since the integral form (2) is to hold for all W . If the density is a constant
(incompressible flow), the equation ∂ρ
∂t +∇ · (ρv) = 0 reduces to ∇ · v = 0. If we suppose, in addition, that the flow is irrotational -that is,∇ × v = 0 or v = ∇u for some potential function u, then we obtain a differential equation (the Laplace equation)
▽2u = 0 for a scalar function u and▽2= ∂2
∂x2 + ∂2
∂y2+ ∂2
∂z2. We shall illustrate how the Laplace transform arises gener- ally in this practical application of the Laplace equation on a half-plane. As we will realize later we don’t have to take the half space as the domain on which the Laplace equation is considered. However, in order to derive the
“general” definition of Laplace transform integrated over the half line we just need to take one of two variable, say x, to be nonnegative.
We consider the Laplace equation of a function u(x, y) in the half-plane (x, y) ∈ R2+ where x ≥ 0 and y ∈ R (i.e., y is a real number) assuming that the function u tends to 0 as√
x2+ y2→ ∞:
▽2u = ∂2u
∂x2 +∂2u
∂y2 = 0.
Now we’ll use the separation of variables method that is a generalization of variation of parameters method in ordinary differential equations by assuming that the so- lutions of the Laplace equation are of the form
u(x, y) = X(x)Y (y)
where X is a function of x alone and Y is a function of y only. It is not hard to show that ▽2u = 0 will have solutions of this form if we can find functions X and Y such that
− (1
X )d2X
dx2 = (1
Y )d2Y
dy2 (3)
for all (x, y) in the half-planeR2+. Noting that the left- hand side of the equation (3) is a function of x only and the right-hand side is a function of y alone, we see that both sides must be equal to a constant, say −t2 with t ≥ 0. Therefore, X and Y will satisfy the following ordinary differential equations
d2X
dx2 = t2X, d2Y
dy2 =−t2Y. (4)
By solving the differential equations of X and Y in (4) above we see that the non-trivial solution
u(x, y) = e−txe−ity = e−tx−ity, t > 0
where eiθ = cos(θ) + i sin(θ). Here we imposed the end behavior of the solution u, that is, u tends to zero as
√x2+ y2→ ∞ and we also took e−ityas the solution to the equation of Y above. For the solution to the differen- tial equation of Y in (4) we are able to think of eityas the solution for it. Then we will have an alternative Fourier transform and its inverse Fourier transform should be the one we are going to define for the Fourier transform in this paper and vice versa. From the theory of Fourier analysis they are just a couple with the transform and inverse transform relationship.
By direct computations we show that for a certain function f (t) the function
u(x, y) =
∫ ∞
0
f (t) e−tx−itydt =
∫ ∞
0
f (t) e−t(x+iy)dt
also satisfies the Laplace equation, ▽2u = 0 in the half- plane. This is the concept of the superposition princi- ple of linear differential equations. Therefore, the solu- tion u to the Laplace equation in the half plane R2+ is just the Laplace transform of a certain function f (t) as the complex-valued function u(x, y) = u(z) where z is a complex number consisting of x > 0 and y as its real and imaginary parts, respectively. Therefore, we finally define the “general” Laplace transform of a function of t, f (t) : [0,∞) → R(or C), as a complex-valued func- tion u(z) with z = (x, y) ∈ R2+ (or on the half complex plane), by an improper integral
u(z) = u(x + iy) =
∫ ∞
0
f (t) e−t(x+iy)dt. (5) Let f (t) = e−ctin (5), for example. Then if you think of u as a real-valued function, i.e., y = 0 then we will have
u(x) = 1 x + c.
It might be very familiar to us because this shows that the “common” Laplace transform of eatwith a constant a is 1
s− a from a table of Elementary Laplace Trans- forms in the class of differential equations you take where the variable s represents the independent variable of the Laplace transformed function.
Now let us extend t in (4) to be any real since we just had said that t is a constant. Then we take X(x) = e−|t|x and Y (y) = e±ity, which shows us that u(x, y) = e−|t|x−ity for any real number t and we, therefore, have
u(x, y) =
∫ ∞
−∞
f (t) e−|t|x−ity dt (6)
for some certain function f (t). Evidently, we should specify the nature of the flowing at the boundary of the domain to obtain a complete description of the physi- cal situation and so we impose the boundary condition u(0, y) = F (y) for a given function F (y) where y is any real. Simply applying the boundary data or x = 0 in (6) shows us that
u(0, y) = F (y) =
∫ ∞
−∞
f (t) e−ity dt (7) or
F (ξ) =
∫ ∞
−∞
f (t) e−iξt dt, ξ is any real, (8) and this is the exact definition of Fourier transform of f (t) as we have seen it in many of the textbooks for mathematical physics. For example, if f (t) = e−|t| then we are able to find its Fourier transform F (ξ) = 2
1 + ξ2as one of exercise problems on this topic. Definitely we also think of the Fourier transform of f (t) as the resulting integral in (6) by choosing the imaginary part of the complex variable z = x + iy or (x, y), i.e., x = 0 as we did for the definition of Laplace transform where we took y = 0.
To summarize both Laplace and Fourier transforms are just the resulting integrals by taking the real and imaginary parts of the complex variable in the solution u(z) = u(x + iy) = u(x, y) to the Laplace equation on the half space (x, y) ∈ R2+ where x ≥ 0 and y ∈ R, respectively.
III. Laplace Transforms and Fourier Transforms on Various Domains
In this section we will present some examples of Laplace transforms of a function f (x) defined on both a finite interval, say (0, 1) for simplicity, and an infinite interval (1,∞).
We consider the Laplace equation of a function u(x, y) in the unit disk centered at the origin on the plane R2. In general finding the temperature u in a solid which is assumed that it is rigid so that the only energy present is thermal energy involves the law of conservation of energy and the Fourier’s law to formulate the Poisson equation or inhomogeneous Laplace equation. When there is no sink or source in the medium the resulting one is called the Laplace equation,▽2u = 0 we actually solved on the half space in the previous section earlier. Thus consid- ering the Laplace equation in the unit disk centered at the origin on the plane R2 is equivalent to finding the steady-state temperature in the unit circular plate lying on the xy plane.
Since we have a circular region it is natural to use polar coordinates and express the Laplace equation as
∂2u
∂r2 +1 r
∂u
∂r +∂2u
∂θ2 = 0, 0 < r < 1 (9) where r is the distance from the origin and θ is the angle measured counterclockwise from the positive x-axis in the polar coordinates. As we did in the previous section we seek separated solutions
u(r, θ) = R(r)Θ(θ). (10) Since (r, θ) and (r, θ + 2π) represent the same point on the plane, it is useful to regard u as 2π periodic in θ;
equivalently, Θ is 2π periodic. For (10) to satisfy the equation (9) we should have
( R′′+1
rR′ )
Θ + 1
r2RΘ′′= 0. (11) Dividing the equation by RΘ/r2we obtain the following equations
R′′+1 rR′− c
r2R = 0, (12)
Θ′′+ cΘ = 0, Θ(θ) = Θ(θ + 2π) (13) where c ≥ 0 is the separation constant (If c < 0, then the differential equation has real exponential solutions, not periodic ones). Thus we have the solutions
Θ = ˜a cos(√
cθ) + ˜b sin(√
cθ) (14)
where ˜a and ˜b are constants, and this function is 2π periodic only when c = n2 for n = 0, 1, 2.· · · . For such n,
Θn= ˜ancos(nθ) + ˜bnsin(nθ) (15) is the solution to (13). If c = n2the equation (12) is an Euler equation which has the general solution
Rn = αnr−n+ βnrn. (16) We must choose αn = 0 to eliminate the singularity at r = 0 and thus we have
Rn= βnrn. (17)
Combining (15) and (17) gives the separated solutions un(r, θ) = rn(ancos(nθ) + bnsin(nθ)), (18) where an and bn are arbitrary constants. Superposing the solutions we just found we obtain
u(r, θ) =
∑∞ n=0
rn(ancos(nθ) + bnsin(nθ)) (19)
As we defined the Laplace transform of a function f in the previous section we can define the Laplace transform of a function f : Z+ → R with its domain of the set of all the non-negative integers,Z+, as
u(r) =L(0,1)(f (n))(r) =
∑∞ n=0
f (n) rn (20)
for 0 < r < 1 by taking θ = 0 or more generally θ = 2kπ for an integer k in u(r, θ) in (19). In other words, the Laplace transform of f :Z+→ R is just a power series of r on the interval (0, 1) such that its coefficients are the function values f (n) for all integers n≥ 0.
Similarly, if we go through all the computation for the Laplace equation in the exterior domain of the unit circle centered at the origin on the plane R2, i.e., r > 1, as we did in previous example then we arrive the following separated solution
u(r, θ) =
∑∞ n=0
(1 r
)n
(ancos(nθ) + bnsin(nθ)) (21)
for 1 < r < ∞ by noting that we have to take βn = 0 in (16) for the decay of the solution at the infinity.
Therefore the Laplace transform of a function f :Z+→ R is defined on (1, ∞) as
u(r) =L(1,∞)(f (n))(r) =
∑∞ n=0
f (n) (1
r )n
. (22)
Surprisingly, its’ just the Laurent expansion of a real- valued function on (1,∞) with the coefficients f(n) for each integer n≥ 0.
In order to define the Fourier transform in this case we assume that the top and bottom of the plate are insulted and that its circumference is kept at a prescribed temper- ature f (θ) and now let us define the Fourier transform of f (θ) by taking r = 1 or imposing a Dirichlet boundary condition u(1, θ) = f (θ) in either the interior or exterior domain as
u(1, θ) = f (θ) =
∑∞ n=0
(ancos(nθ) + bnsin(nθ)) (23)
and this trigonometric series is the exact definition of Fourier series of a periodic function f (θ) on the unit circle where an and bn are the Fourier coefficients of f (θ) as we might expect in the first place. Very interestingly, the Fourier transform of a periodic function in the interval (0, 2π) is just the Fourier series expansion of the function that is one of the most important topics dealing with problems arising from mathematical physics.
IV. Conclusion
As pointed out the approach taken and adopted by many textbooks in both mathematical physics and even mathematics to deal with integral transforms such as Laplace and Fourier transforms might make the students confused by their definitions because they look rather strange and demotivating as starting with improper in- tegrals out of nowhere. Students also might find it dif- ficult to understand what they are doing when they use those integral transforms to solve application problems in their majors.
Therefore, in this article, we present an alternative approach to derive the “general” definitions of integral transforms on the complex plane with the example of the Laplace equation on the half space in potential theory. In a summary both Laplace and Fourier transforms are just
the resulting integrals by taking the real and imaginary parts of the complex variable in the solution u(z) = u(x+
iy) = u(x, y) to the Laplace equation on the half space (x, y)∈ R2+ where x≥ 0 and y ∈ R, respectively. Then we help students understand the Laplace and Fourier transforms on various domains such as the unit interval, (0, 1) and (1,∞) by solving a boundary value of problem of Laplace equation on the unit disk.
Specifically, motivated by the general definition of the Laplace transform on the half-complex plane we can de- fine two different Laplace transforms,L(0,1) andL(1,∞), on both a finite length of an interval, (0, 1), and an infi- nite interval, (1,∞). As we already noticed in the pre- vious examples these are just the power and Laurent ex- pansions of a variable r, respectively. One of applications of the Laplace transform on (0,∞) we usually teach in the courses of differential equations for the undergrad- uates is to solve initial value problems by transforming differential equations into corresponding algebraic equa- tions where we apply the integration by parts for terms involving the derivatives. Hence a natural question now arises. How do these Laplace transforms we defined dif- ferently provide solutions to differential equations that can not be solved by more elementary means? The an- swer should be simple as we might already know, that is, it’s the power series method. In other words, we seek to find the solution in the form of the power series by identifying all the coefficients in it. Also it should be noted that there are infinitely many ways to choose the angle θ, not only θ = 0, to define the Laplace trans- forms on those domains. If you take the angle θ to be different angle from the x-axis then we will have a new Laplace transform containing Fourier series expansion of a periodic function.
For the Fourier transforms of functions on the unit circle or periodic functions in the interval (0, 2π) we re- alize from the examples in the previous section that it’s just the Fourier series expansion of the periodic function and hence the Fourier transform on the entire domain can be thought of as the extension of the Fourier series of periodic functions on a finite interval. Even though we have the case of an infinite interval with a boundary point, say, (1,∞) we still have the same Fourier series expansion due to the boundary condition at r = 1 as expected.
The most significant benefit in this paper is that the presentation engages the students in the constructive and discoverable ways to understand the Laplace and Fourier transforms rather than simply giving them the definition as an improper integral and have them use those trans- forms computationally and procedurally. Also introduc- ing the Law of Conservation of Mass and Laplace equa- tions in potential theory to define those integral trans- forms makes them accessible and motivating to the stu- dents, in particular, the physics majors to engage in the study of mathematical physics effectively and efficiently.
As pointed out in the Introduction Laplace and Fourier transforms play an immensely important role in many applications such as electrodynamics (wave equations), optics (anharmonic waves), engineering (signal process- ing), and quantum mechanics (Schrodinger equation).
Hence we hope that this article has helped the reader to study integral transform methods in solving differen- tial equations and to get tools for use even in a variety of boundary value problems.
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