In this paper we introduce Coleff–Herrera residue currents defined by systems of c-holomorphic functions and prove a Lelong–Poincaré and a Cauchy-type formula as well as the transformation law for these currents.
"Residue calculus for c-holomorphic functions." Ark. Mat. 47 (1) 73 - 89, April 2009. https://doi.org/10.1007/s11512-008-0088-7