Identities for the E2,1-transform and their applications

doi:10.1016/j.amc.2006.09.139

Applied Mathematics and Computation

Volume 187, Issue 2, 15 April 2007, Pages 1557-1566

https://doi.org/10.1016/j.amc.2006.09.139 Get rights and content

Abstract

In the present paper the authors introduce the $E_{2, 1}$ -transform with kernel the exponential integral function. It is shown that the third iterate of the $L_{2}$ -transform is the exponential integral transform, and some identities involving the new transform, the $L_{2}$ -transform and the Widder potential transform are given. Using the identities, Parseval–Goldstein type results involving these transforms are proved. Some illustrative examples are also given.

Introduction

In this paper, we introduce the $E_{2, 1}$ -transform as $E_{2, 1} {f (x); y} = \int_{0}^{\infty} x \exp (x^{2} y^{2}) E_{1} (x^{2} y^{2}) f (x) d x,$ where E₁(x) is the exponential integral function defined as $E_{1} (x) = - Ei (- x) = \int_{x}^{\infty} \frac{e^{- u}}{u} d u = \int_{1}^{\infty} \frac{e^{- xt}}{t} d t .$ The $L_{2}$ -transform $L_{2} {f (x); y} = \int_{0}^{\infty} x \exp (- x^{2} y^{2}) f (x) d x,$ is introduced in [12] and the Widder potential transform $P {f (x); y} = \int_{0}^{\infty} \frac{xf (x)}{x^{2} + y^{2}} d x,$ is studied in [6]. It is well known that the second iterate of the $L_{2}$ -transform is the Widder potential transform; that is, $L_{2}^{2} {f (x); y} = L_{2} {L_{2} {f (x); u}; y} = P {f (x); y},$ provided that integrals involved converge absolutely, (cf. [12, Eq. (2.1)]). These transforms were studied in [7], [8], [10], [11].

The $L_{2}$ -transform and the Laplace transform are related by the identity $L_{2} {f (x); y} = \frac{1}{2} L {f (\sqrt{x}); y^{2}},$ in which the Laplace transform is defined as $L {f (x); y} = \int_{0}^{\infty} \exp (- xy) f (x) d x .$ The Widder potential transform and the Stieltjes transform are related by the identity $P {f (x); y} = \frac{1}{2} S {f (\sqrt{x}); y^{2}},$ in which the Stieltjes transform is defined as $S {f (x); y} = \int_{0}^{\infty} \frac{f (x)}{x + y} d x .$ In this paper, it is shown that the third iterate of the $L_{2}$ -transform is a constant multiple of the $E_{2, 1}$ -transform as defined in (1.1). Using the identities, Parseval–Goldstein type theorems relating these integral transforms are proved. As an application of the identities and theorems, some illustrative examples are given.

Section snippets

The main theorem

In the following lemma, we show that the third iterate of the $L_{2}$ -transform is essentially the $E_{2, 1}$ -transform.

Lemma 1

The identities $L_{2}^{3} {f (x); y} = \frac{1}{2} P {L_{2} {f (x); u}; y} = \frac{1}{2} L_{2} {P {f (x); u}; y},$ and $L_{2}^{3} {f (x); y} = \frac{1}{4} E_{2, 1} {f (x); y},$ hold true, provided that the integrals involved converge absolutely.

Proof

The proof of identity (2.1) easily follows from (1.5). In order to prove (2.2), we use (2.1) $L_{2}^{3} {f (x); y} = \frac{1}{2} L_{2} {P {f (x); u}; y} .$ Using the definitions of the $L_{2}$ -transform (1.3) and the Widder potential transform (1.4) we have $L_{2}^{3} {f (x); y} = 1$

Useful corollaries

Interesting consequences of the main theorem will be given in this section. Useful Parseval-type identities for the $L_{2}$ -transform and the Widder potential transform are contained in

Corollary 6

If the integrals involved converge absolutely, then we have $\int_{0}^{\infty} y^{ν} L_{2} {f (x); y} d y = \frac{1}{2} Γ (\frac{ν + 1}{2}) \int_{0}^{\infty} \frac{f (x)}{x^{ν}} d x,$ where $R (ν) > - 1$ , $\int_{0}^{\infty} \frac{P \{g (u); y\}}{y^{ν}} d y = \frac{π}{2} \sec (\frac{ν π}{2}) \int_{0}^{\infty} \frac{g (u)}{u^{ν}} d u,$ where $- 1 < R (ν) < 1$ , and $\int_{0}^{\infty} \frac{P {f (x); y}}{y^{ν}} d y = Γ (\frac{1}{2} - \frac{ν}{2}) \int_{0}^{\infty} y^{ν} L_{2} {f (x); y} d y,$ where $R (ν) < 1$ .

Proof

We set $g (u) = u^{ν - 1}$ in relation (2.31) of Theorem 5, then use (3.4), (2.8), (2.10) in the

Illustrative examples

An illustration of the Parseval–Goldstein type relation (3.1) of Corollary 6 is given in the following example.

Example 13

We have $\int_{0}^{\infty} y^{μ - 1} \exp (a^{2} y^{2}) Erfc (ay) d y = \frac{a^{- μ}}{2} Γ (\frac{μ}{2}) \sec (\frac{π μ}{2}),$ where $0 < R (μ) < 1$ .

Proof

We set $f (x) = \frac{1}{\sqrt{x^{2} + a^{2}}}$ in relation (3.2) of Corollary 6, then using relation (1.6) and the formula [1, Entry 18, p. 135], we have $L_{2} {f (x); y} = \frac{1}{2} L \{(x + a^{2})^{- 1 / 2}; y^{2}\} = \frac{\sqrt{π}}{2 y} \exp (a^{2} y^{2}) Erfc (ay) .$ Substituting (4.2), (4.3) into (3.2) of Corollary 6, we find $\int_{0}^{\infty} y^{μ - 1} \exp (a^{2} y^{2}) Erfc (ay) d y = \frac{1}{\sqrt{π}} Γ (\frac{μ}{2} + \frac{1}{2}) \int_{0}^{\infty} \frac{x^{- μ}}{\sqrt{x^{2} + a^{2}}} d x .$ The integral on the right-hand side has the

References (12)

O. Yürekli
New identities Involving the Laplace and the $L_{2}$ -transforms
Appl. Math. Comput.
(1999)
A. Erdelyi
(1954)
A. Erdelyi
(1954)
M.L. Glasser
Some Bessel function integrals
Kyungpook Math. J.
(1973)
N. Lebedev
Special Functions and their Applications
(1965)
H.M. Srivastava et al.
A theorem on Stieltjes-type integral transform and its applications
Complex Var. Theory Appl.
(1995)

There are more references available in the full text version of this article.

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Identities for the E2,1-transform and their applications

Abstract

Introduction

Section snippets

The main theorem

Useful corollaries

Illustrative examples

Appl. Math. Comput.

Some Bessel function integrals

Kyungpook Math. J.

Special Functions and their Applications

A theorem on Stieltjes-type integral transform and its applications

Complex Var. Theory Appl.

Identities for the $E_{2, 1}$ -transform and their applications