Integral representations for products of two solutions of the Airy equation with shifted arguments and their applications in physics

Integral representations for a complete set of linearly independent products of two solutions of the Airy equation whose arguments differ by \(z_0\) $z_0$ are obtained using the Laplace contour integral method. This generalizes similar integral representations for the case \(z_0=0\) $z_0=0$ obtained by Reid. The relation to other previous results is discussed. The results are used to obtain the outgoing-wave Green’s function for an electron in a static electric field in a closed analytic form.