In class, I saw Green's Theorem defined as follows:
Proposition Let $\Omega$ bounded open subset of $\mathbb{C}$, assume $\partial \Omega$ can be parameterized by piecewise $C^\infty$ curves. $f:\overline \Omega \to \mathbb{C}$ a $C^1$ function, write $f=u+vi$ where $u,v:\overline{\Omega}\to \mathbb{R}$ $C^1$ functions. Then $$\int_{\partial \Omega}f(z)dz=2i\int_{\Omega}\frac{\partial f}{\partial \overline z}dA$$ with positive orientation.
In another textbook (Schaum's Outline of Complex Variables) I saw Green's Theorem written as follows:
4.10 Complex Form of Green's Theorem Let $F(z, \bar{z})$ be continuous and have continuous partial derivatives in a region $\mathcal{R}$ and on its boundary $C$, where $z=x+i y, \bar{z}=x-i y$ are complex conjugate coordinates [see page 7]. Then Green's theorem can be written in the complex form $$ \oint_{C} F(z, \bar{z}) d z=2 i \iint_{\mathcal{R}} \frac{\partial F}{\partial \bar{z}} d A $$ where $d A$ represents the element of area $d x d y$.
I am trying to relate these two definitions. Why in the second version, does the function $F$ take arguments $z$ and $\overline z$ while the first only takes argument the argument $z$?