I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider f=u-iv as a conservative vector field, mapping $C-> C$.
But what can we say about f=u+iv? Is it also conservative?
I can't verify that it is also curl-free, using the C-R equations.
Thanks,