Complex contact and lift transformations

We study mappings from sets of real variables into complex variables, which extend features of lift and contact transformations between real variables that we explored in a previous paper. In particular the relationship between lifts in R2n+1R2n+1 and the Cauchy–Riemann equations for functions of nn complex variables is discussed. Explicit examples are given to illustrate the anatomy of such transformations, including the occurrence of singularities. Applications to nonlinear partial differential equations arising in fluid mechanics are presented.