Extended Newman-Janis algorithm for rotating and Kerr-Newman de Sitter and anti de Sitter metrics

The Newman-Janis algorithm is well known to provide rotating black holes solutions to Einstein's equations from static seeds, through a complexification of a radial and a time coordinates. However, an ambiguity remains for the replacement of the $r^{-1}$ and $r^{-2}$ powers of the radial coordinate. We show here that the two cases are unified by a simple expression which allows its extension to the $r^{2}$ power, characteristic of the de Sitter ($dS$) and anti de Sitter ($AdS$) spacetimes. The formula leads almost automatically to the Kerr and Kerr-Newman-$dS$ and -$AdS$ metrics.