Analytic Expressions for Mean Magnification by a Quadrupole Gravitational Lens

We derive an analytic expression for the mean magnification due to strong gravitational lensing, using a simple lens model, a singular isothermal sphere embedded in an external shear field. We compute separate expressions for 2-image and 4-image lensing. For 4-image lensing, the mean magnification takes a particularly simple form: ₄ = 3. 6/ (gamma* (1-gamma²) ), where gamma is the external shear. We compare our analytic results to a numerical evaluation of the full magnification distribution. The results can be used to understand the magnification bias that favors the discovery of four-image systems over two-image systems in flux-limited lens surveys.