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 z0 are obtained using the Laplace contour integral method. This generalizes similar integral representations for the case z0=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.