Identities for the exponential integral and the complementary error transforms
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.
scientific article; zbMATH DE number 5115409