I'm reviewing complex analysis for the GRE. I've never taken a course in complex analysis before, but I do know vector calculus.
I'm trying to understand the statement of the complex version of Green's theorem, which has a few symbols that I've never seen before.
$$ \oint_C F(z,\overline{z}) dz = 2i \iint_{R} \frac{\partial F}{\partial \overline{z}} dA $$
For example, what is $F(z,\overline{z})?$ How is $\partial F/\partial \overline{z}?$ defined?
Also, if there is an obvious connection to Green's theorem on a plane (the real version) I would appreciate an explanation!