The complex theory of gravitational lensing: Beltrami equation and cluster lensing

We introduce a new complex formalism to describe arclet fields in clusters of galaxies and derive the appropriate inversion techniques to find the related mass distribution of the lensing cluster. Applying the complex formalism to the statistics of lensed background sources shows that the mean of the axial ratios of locally lensed isotropically distributed elliptical sources is {\em not} influenced by the intrinsic ellipticity distribution, if the cluster is subcritical in that region.