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July 2016 | Vol. 19 No. 2

AUP Research Journal | ISSN 1655-5619 80

Theorems on nth Dimensional Laplace TransformMichael Sta. Brigida1, Edwin Balila2, Edna Mercado3

1Cavite State University2Adventist University of the Philippines

3De La Salle University - Dasmariñas

Abstract

Let U be the set of all functions from [0,∞)n to and V be the set of all functions from S C n to . Then the nth dimentional Laplace transform is the mapping

Ln : U → Vdefined by:

Where In this paper we gave alternative proof for some theorems on properties of nth dimensional Laplace Transform, we proved that if is piecewise continuous on [0,∞)n and function of exponential order then the nth dimensional Laplace transform defined above exists, absolutely and uniformly convergent, analytic and infinitely differentiable on Re (s1) > 1, Re (s2) > 2, ..., Re (s2) > n, and we gave also some corollaries of these results.

Keywords: theorems, Laplace transform

Laplace transform is a widely used integral trans-form with many applications in Physics, Engineer-ing and Applied Mathematics. It is named after Pierre-Simon Laplace, who introduced the trans-form in his work on probability theory. It is also very useful for solving ordinary and partial differ-ential equations, difference equations, integral equations as well as mixed equations. In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and me-chanical sytems. It can be used as a very effective tool in simplifying the calculations in many fields of engineering and mathematics. It also provides a powerful method for analyzing linear systems.

Estrin and Higgins [21] in 1951 gave the systematic account of the general theory of an operational calculus using double Laplace trans-form and illustrated its application through solu-

tion of two problems, one in electrostatics, the other in heat conduction. Coon and Bernstein [22] conducted systematic study of double La-place transform in 1953 and developed its prop-erties including conditions for transforming de-rivatives, integrals, and convolution. Buschman [23] in 1983 used double Laplace transform to solve a problem on heat transfer between a plate and a fluid flowing across the plate. In 1989, Debnath and Dahiya [11], [12] established a set of new theorems concerning multidimensional Laplace transform of some functions and used the technique developed to solve an electrostat-ic potential problem. In 1992, Brychov, Glaeske, Prudnikov, and Tuan [27] in their book entitled Multidimensional Integral Transformations dis-cussed and proved several theorems on the properties of nth dimensional Laplace transform. In 2013, Aghili and Zeinali [24] implemented mul-

Vol. 19 No. 2 | July 2016

AUP Research Journal | ISSN 1655-561981

tidimensional Laplace transform method for solv-ing certain non-homogeneous forth order partial differential equations. Atangana [26] discussed some properties of triple Laplace transform and applied to some kind of third order partial dier-ential equations. In 2014, Eltayeb, Kilicman, and Mesloub [18] applied double Laplace transform method to evaluate the exact value of double innite series. Hamood Ur Rehman, Muzammal Iftikhar, Shoaib Saleem, Muhammad Younis, and Abdul Mueed [19] discussed some properties and theorems about the quadruple Laplace transform and gave a good strategy for solving the fourth order partial differential equations in engineer-ing and physics by quadruple Laplace transform. Dhunde and Waghmare [20] presented the ab-solute convergence and uniform convergence of double Laplace transform and solved a Volterra Integro Partial Differential Equation using double Laplace transform.

Research ObjectiveThe main objective of this paper was to

extend some results in [10], [19], [20] [24] and [26], provide alternative proof for some results in [27] on theorems on the properties of the nth dimensional Laplace transform and derive some special multiple improper integral identities.

THE nth DIMENSIONAL LAPLACE TRANS-FORM

In this section we will deal with the main results. For brevity, we will use the following no-tation throughout this chapter.

Definiftion 1. Let U be the set of all func-tions from [0, )n to and V be the set of all functions from S to . Then the nth dimen-sional Laplace transform is the mapping

defined by:

Where

From the above definition, it is clear that

Occasionally, we also denote

It is also clear that there are some func-tions in U in which the nth dimensional Laplace transform does not exist. We now define a class of functions needed for the existence of the nth dimensional Laplace transform.

Definition 2. A function is called piecewise continous in the region if the region can be divided into a finite number of sub-region such that is continuous.

Definition 2 is a natural extension to nth

dimension of piecewise continuity and from this definition we can evaluate the nth dimentional La-place Transform using theorems on multiple in-tergrals. We now define other class of functions needed for the existence of nth dimensional La-place transform.

Definition 3. If positive real constant M and a vector in exist such that at least one i of xi > Ni is true for i = 1, 2, 3..., n we have:

then is said to be function of exponential order Definition 3 is an extension to nth dimen-sion for the functions of exponential order. Func-tions of exponential order cannot grow in abso-lute value more rapidly than if at least one of increases. Clearly, the exponential function

has exponential order =

July 2016 | Vol. 19 No. 2


(a1, a2, ..., an), whereas has exponen-tial order = where at least one of > 0 and any Bounded functions like sin , cos

, tan-1 have exponential order = (0, 0, ..., 0) where and

are continuous functions from to while has order = (-1, -1, ..., -1).

However, does not have expo-nential order.

CONVERGENCE THEOREM The following theorem gives us sufficient conditions for the existence of nth dimensional La-place transform. Theorem 1. If is piecewise contin-uous in every bounded region and of exponential order then the nth dimensional Laplace trans-form exists for all Re(s1) > , Re(s2) > ,...., Re(sn) > . Proof: From Definition 1 we have =

Hence we get:

where is a bound-ed region, is an unbounded region such that

and Clearly ,the integral

exists, since is piecewise continuous in ev-ery bounded region.So, we only need to show that the integral

exists.Now, we have a chain of inequality:

but since is of exponential order

where now the integral:

is equal to

Hence, if

then:

Therefore, if exists.

We now prove the absolute convergence of the nth dimensional Laplace integral.

Theorem 2. If is piecewise contin-uous on [0, )n and of exponential order then the nth dimensional Laplace transform:

is absolutely convergent for

Proof:From Theorem 1

if

thus

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Theorems on nth Dimensional Laplace Transform

is absolutely convergent for all

Similar result for Theorem 2 can be found in [22] where the function is considered a map-ping from [0, )n to while in this result we con-sidered our function to be a mapping [0, )n to with additional condition that the function

must be piecewise continuous on [0, )n. In this result, we actually found the bound for the integral

We now prove two lemmas needed for the proof of the uniform convergence of the nth dimension-al Laplace transform.

Lemma 1. If is piecewise continu-ous on [0, )n and of exponential order such that , then

is piecewise continuous on [0, )n 1and of exponential order , Where are n - 1 dimensional vectors without and .

Proof:The function

is piecewise continuous on [0, )n, since is piecewise continuous on [0, )n, so it follows that is also piecewise continuous on [0, )n 1. we need only to show that is a function of exponential order , now

Hence

But it is clear that

is less than

Therefore,

is less than

for at least one of 1, 2, 3, ..., nthus,

That is, is of exponential order

Lemma 2. If is piecewise continuous on [0, )n and of exponential order then

is uniformly convergent on for

Proof:We know that

and from Lemma 1 we have:

provided for 1, 2, ..., n.whenever

then

July 2016 | Vol. 19 No. 2


So, we choose

,

this N is always positive for that is given any , there exists a value N > 0 such that

whenever for all values of with

This is precisely the condition required for the uniform convergence in the region

The following theorem proves the uni-form convergence of the nth dimensional Laplace transform.

Theorem 3. If is piecewise contin-uous on [0, )n and of exponential order then

converges uniformly on

Proof:Without loss of generality, write as

where

now by Lemma 2,

converges uniformly on Hence is piecewise continuous, by Lemma 1, on [0, )n-1 and of exponential order

so again by Lemma 2 the integral


Thus, by applying Lemma 1 and Lemma 2 repeat-edly we see that the intergral

is uniformly convergent on

Similar result for Theorem 3 can be found in [27]. In our result, we used two lemmas to prove the uniform convergence of the nth dimentional La-place integral.

In what follows, we will state that based on Theorem 3, the order limit and integral can safely be reversed. The proof is straightforward.

COROLLARY 1. If is piecewise con-tinuous on [0, )n and exponential order then

is continuous and integrable complex valued func-tion in real variables on

Other implication of Theorem 3 is again stated as corollary.

COROLLARY 2. If the integral is convergent, then

converges uniformly on for

BEHAVIOR OF THE nth DIMENSIONAL LAPLACE TRANSFORM

From Corollary 1, we can insert the limit on any point on the region

inside the nth dimensional Laplace integral and thus

can be expressed as

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Vol. 19 No. 2 | July 2016


which can be further simplified as

ANALYTICITY OF THE nth DIMENSIONALLAPLACE TRANSFORM Analyticity plays an important role in

complex analysis and in physics, since once the function was determined to be analytic in some region then its higher order derivatives are also analytic in the same region. Also the real and imaginary parts of the function are definitely many times differentiable and harmonic in the same region.

Before proving the analyticity of nth di-mensional Laplace transform we first prove the following lemmas.

LEMMA 3. If is piecewise continu-ous on [0, )n and of exponential order then the integrals

converges uniformly on , where

Proof of (a):By direct evaluation we get

for But the integral

converges for Thus, by Weiertrass M test, the integral


Proof of (b): The proof is analogous to the proof of (a).

We now prove the analyticity of nth dimensional Laplace transform

Theorem 4. If is piecewise contin-uous on [0, )n and of exponential order then

is analytic function on

Proof: Let , then

where and are re-spectively expressed as

and

By Lemma 3, we can reverse the order of derivative and the integrals of the real and imagi-nary parts of and hence the integrals

where

and

and uniformly convergent on

Thus,

are continuous on for

July 2016 | Vol. 19 No. 2


and also, we see that:

and

that is, Cauchy-Riemann equations are satisfied on for Thus,

is analytic function on

Similar result for Theorem 4 can be found in [27]. In our result, we used Lemma 3 to prove the analyticity of the nth dimensional Laplace transform. From Theorem 4 it follows that the nth dimensional Laplace transform is infinitely many times differentiable and continuous on

and hence can be differentiated of all orders in-side the nth dimensional Laplace integral operator.

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