On a two-dimension infinite plane we can always denote a complex number z which satisfies: $$ z=x+iy\\ \bar{z}=x-iy $$ and write down the surface element $dxdy$ as $\frac{1}{2}dzd\bar{z}$..Then my first question is when I meet a integral such as: $$ I=\int f(z,\bar{z})dzd\bar{z} $$ I have no idea how to evaluate it directly. All I can do is to transform it back to the variables (x,y) and do it in an ordinary method. I wonder what is the geometric meaning of $dzd\bar{z}$ and can I do the integral in a direct way?
Also, If we make a variables transformation: $$ z \rightarrow \omega(z,\bar{z}) \\ \bar{z} \rightarrow \bar{\omega}(z,\bar{z}) $$ Is it still correct that $$ I=\int f(z(w,\bar{w}),\bar{z}(w,\bar{w}))\frac{(\partial z,\partial \bar{z})}{(\partial \omega, \partial \bar{\omega})}d\omega d\bar{\omega} $$ Where the Jacobi factor is same to one in a real variable transformation.
Thanks a lot!