A four parameter integral identity and a few consequences

Abstract

The identity

\[
\int_0^\infty \frac{e^{-ax} \, e^{-b\sqrt{x^2 + 2cx + a^2}}}{\sqrt{x^2 + 2cx + b^2}} \, dx
=
\int_0^\infty \frac{e^{-bx} \, e^{-a\sqrt{x^2 + 2cx + b^2}}}{\sqrt{x^2 + 2cx + a^2}} \, dx
\]

is derived, applied to the Struve function $H_0$, and used to deduce the reduction formula

\[
\int_0^\infty \frac{F(\sqrt{x^2 + 2cx + b^2} + x)}{\sqrt{x^2 + 2cx + b^2}} \, dx
=
\int_0^\infty f(t)e^{-t} E_1[(c+b)t] \, dt,
\]

where $F$ is arbitrary.