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: <mu>_4 = 3.6/(gamma*(1-gamma^2)), 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. 
Tehani K. Finch (Howard U.), Lisa P. Carlivati (Harvard U.), Joshua N.   Winn (Harvard-Smithsonian CFA), Paul L. Schechter (MIT)