Let $\displaystyle D = \mathbb{C} - \{0\} $.

Is there a function $\displaystyle F: D \rightarrow \mathbb{C} $

such that $\displaystyle d/dz(F(z)) = 1/z $ $\displaystyle \forall z \in D$?

Similarly, is there a function $\displaystyle F: \mathbb{C} \rightarrow \mathbb{C} $

such that such that $\displaystyle d/dz(F(z)) = \overline{z} $ $\displaystyle \forall z \in \mathbb{C}$?

Is this as simple as integrating? In class, we have only covered what it means when a complex function is integrated over a path. What such path should be used? Or is there such a thing as a "pathless" complex integral?

Thanks!