surface integrals in the complex plane in polar form

Question

if i am not mistaken the definition of a 1d integral in the complex plane of some analytical $f(z)=u+i v$ where $z=x+iy=re^{i\theta}$ is

$\int\limits_\gamma f(z)dz=\int\limits_\gamma udx-vdy+i\int\limits_\gamma udy+vdx$

i thought that a surface integral then would be $\int f dxdy=\int f Re(dz)Im(dz)$

and i was told that it is the same as $\int f dz(dz)^*$

however $(dz)^*$ is clearly not $dx-idy$

and i am not sure what it should be

my question is:

how do i go from $\int dz(dz)^*$ to $\int drd\theta$