The physical optics fields of an aperture on a perfectly conducting screen in terms of line integrals

The physical optics fields due to an aperture on a perfectly conducting screen are expressed in terms of line integrals over the boundary of the aperture. These line-integral representations have the same properties as the ones in terms of surface integrals over the aperture; in particular, they admit arbitrary, source distributions, and are continuous everywhere in the source-free half-space, including the geometric optics shadow boundary.