  We propose a definition of an exact lens equation without reference to a
background spacetime, and construct the exact lens equation explicitly in the
case of Schwarzschild spacetime. For the Schwarzschild case, we give exact
expressions for the angular-diameter distance to the sources as well as for the
magnification factor and time of arrival of the images. We compare the exact
lens equation with the standard lens equation, derived under the
thin-lens-weak-field assumption (where the light rays are geodesics of the
background with sharp bending in the lens plane, and the gravitational field is
weak), and verify the fact that the standard weak-field thin-lens equation is
inadequate at small impact parameter. We show that the second-order correction
to the weak-field thin-lens equation is inaccurate as well. Finally, we compare
the exact lens equation with the recently proposed strong-field thin-lens
equation, obtained under the assumption of straight paths but without the small
angle approximation, i.e., with allowed large bending angles. We show that the
strong-field thin-lens equation is remarkably accurate, even for lightrays that
take several turns around the lens before reaching the observer.
