On the usual approximation used in the Rayleigh–Sommerfeld diffraction theory

The complete Rayleigh–Sommerfeld scalar diffraction formula contains (1-ikR) in the integrand. Usually the wavelength is small compared with the distance of the observation point from the aperture and (1-ikR) is approximated by -ikR alone. Other approximations usually made in the Rayleigh–Sommerfeld formula are addressed as well. Closed-form solutions, without approximations, are possible wherein interesting consequences of these approximations become apparent.