# Mapping and Cauchy- Reimann conditions

If a complex function is analytic it must hold Cauchy-Reimann Conditions, So conjugate of F(Z) is irrotational and solenoidal , Does this points mapping from $$R^2$$ to a subspace of $$R^2$$ ($$R^1$$) , if F(Z) maps to $$R^2$$ we dont get a divergent less field , But the above interpretation does not satisfy idea of conformal mappings , can any one correct me?