**Bulletin of Taras Shevchenko National University of Kyiv. Astronomy, no. 63, p. 21-32 (2021)**

**Complex formalism in the theory of gravitational lensing and the geometry of caustics**

** A. Alexandrov, PhD, Senior Researcher
**

**Taras Shevchenko National University of Kyiv**

**Abstract**

In the theory of gravitational lensing, the critical curves and caustics of the lens mapping are of particular interest. The most striking features of this phenomenon take place just when the source is in the vicinity of the caustic of the gravitational lens system. The main method for studying the properties of a lens mapping in the vicinity of a critical point is its approximation by a segment of a Taylor series in a special local coordinate system; this allows one to describe all the necessary properties with sufficient accuracy by means of a certain number of Taylor coefficients. In this article, we propose a general algorithm for calculating the mentioned coefficients directly in the original coordinates. The algorithm essentially uses the complex formulation of the lensing equations and the parameterization of critical curves, which was first proposed by Witt (1990). We analyzed the formula for the curvature of the caustic and introduced a closely related function D. For D > 0, the so-called positive side of the caustic neighborhood, whose points have two more images, corresponds to the convexity of the caustic, and for D < 0, to its concavity. The critical points, at which D = 0 , correspond to the inflection points of the caustic. The conditions for the critical point to be a cusp, as well as the positivity and negativity of the cusps are considered. The properties of caustics are illustrated with examples of the Chang-Refsdal lens and a simplified dark matter clump model.

**Key words
**Gravitational lensing, complex formulation, critical curves and caustics.

**References**