'Flux conservation' by a Schwarzschild gravitational lens

It is demonstrated, by an explicit calculation, that a single, isolated Schwarzschild gravitational lens leads to a vanishing net total amplification on its own, without any 'help' from other lenses. It is shown that, in the limit of small M/R (where M is the mass of the lens in geometrical units and R is the distance from the source to the observers), the positive net total amplification, obtained from the classical approximation of the amplification for small impact angle lensing, is balanced by a negative net total amplification from the first-order term in M/R in an approximation of the amplification for large impact angle lensing derived here. The relation of these results to the common interpretation of average flux conservation by gravitational lenses is discussed as due to the collective action of many lenses in a cosmological framework.