Magnification relations in gravitational lensing via multidimensional residue integrals

Gravitational lensing has proven to be not only important astrophysically, but intriguing mathematically as well. Mathematical investigations of gravitational lens theory have yielded important results and insights, employing techniques and results from such disparate areas as catastrophe theory, differential geometry and Morse theory. In this paper, we illustrate how another seemingly unrelated subject, multidimensional residue calculus, applies to gravitational lensing, and specifically we explain the origin of certain “magnification relations” that have been discussed in the lensing literature. We additionally demonstrate that calculations of these magnification relations are enormously simplified using residue techniques, and illustrate the method by deriving several new results. This paper is organized as follows. In the remainder of this section, we introduce the relevant terminology of gravitational lensing, and describe the magnification relations. In Section II, we express the problem in terms of residue calculations at the image positions, and thereby relate it to a residue at infinity. Using one-dimensional residue calculus, we derive trace formulas for the magnification relations for a subset of lens models. In Section III, we consider the general class of lens models, and describe how to perform the necessary multidimensional residue integrals. In Section IV, we apply this formalism and derive previously known results, as well as new results. In Section V, we summarize our results and discuss implications. The material discussed in sections II-IV may be unfamiliar to astronomers, and so we provide a simple procedure which may be applied to models to obtain their magnification relations, without requiring a detailed understanding of the underlying mathematics.