  In this paper we introduce the two possible formulations of the F-functional
calculus which are based on the Fueter-Sce mapping theorem in integral form and
we introduce the pseudo F-resolvent equation. In the case of dimension 3 we
prove the F-resolvent equation and we study the analogue of the Riesz
projectors associated with this calculus. The case of dimension 3 is also
useful to study the quaternionic version of the F-functional calculus.
