A four parameter integral identity and a few consequences

Abstract

The identity \[ \int_0^\infty e^{-\alpha x}\frac{e^{-a\sqrt{x^2+2\beta x+b^2}}}{\sqrt{x^2+2\beta x+b^2}} \, dx = \int_0^\infty e^{-\beta x}\frac{e^{-b\sqrt{x^2+2\alpha x+a^2}}}{\sqrt{x^2+2\alpha x+a^2}} \, dx \] is derived, applied to the Struve function \(\mathbf{H_0}\) and used to deduce the reduction formula \[ \int_0^\infty \frac{F(\sqrt{x^2+2\beta x+b^2}+x)}{\sqrt{x^2+2\beta x+b^2}} \, dx = \int_0^\infty f(t)e^{\beta t} E_1[(\beta+b)t] \, dt, \] where \(F\) is arbitrary.