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

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