Identities for the exponential integral and the complementary error transforms
type: Item
instance of: scholarly article
title: Identities for the exponential integral and the complementary error transforms
zbMATH Open document ID: 1109.44003
author: N. E. Zubov
published in: Applied Mathematics and Computation
publication date: 2007-01-11T00:00:00Z
review text: The authors introduce two new integral transforms, the exponential integral transform \[ \mathcal E_1\{ f(x);y\{=\int_0^{\infty}\exp(xy)E_1(xy)f(x)\,dx\tag{1} \] where \(E_1(x)\) is the exponential integral function defined as \[ E_1(x)=-Ei(-x)=\int_x^{\infty}\frac{e^{-x}}x \,dx \] and the complementary error integral transform \[ \mathcal E_c\{f(x);y\}=\int_0^{\infty}\exp(xy) Erfc(\sqrt{xy})f(x)\,dx.\tag{2} \] It is shown that the third iterate of the Laplace transform is the exponential integral transform as defined in (1) and a modified third iterate of the Laplace transform is the complementary error transform (2). Some identities involving these new transforms, the Laplace transforms and the Stieltjes transforms are given. Using these identities, Parseval-Goldstein type results involving these transforms are proved. Few examples are also given.
reviewed by: R. S. Dahiya
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 44A10
zbMATH DE Number: 5115409
zbMATH Keywords: exponential integral tranforms
zbMATH Keywords: Laplace transforms
zbMATH Keywords: Stieltjes transforms
zbMATH Keywords: Parseval-Goldstein type theorems
zbMATH Keywords: complementary error integral transform
MaRDI profile type: Publication
full work available at URL: j.amc.2006.05.024
OpenAlex ID: W2008484319
cites work: Q5827353
cites work: A theorem on Widder's potential transform and its applications
DOI: 10.1016/J.AMC.2006.05.024
description: scientific article; zbMATH DE number 5115409