Show that the Cauchy-Riemann equations for $f: \mathbb{C} \rightarrow \mathbb{C}$ are equivalent to

$$\frac{\partial f}{\partial \bar{z}}=0 \text {, }$$

where $z=x+i y$, and $\partial / \partial \bar{z}$ should be defined in terms of $\partial / \partial x$ and $\partial / \partial y$. Use Green's theorem, together with the formula $d z d \bar{z}=-2 i d x d y$, to establish the generalised Cauchy formula

$$\oint_{\gamma} f(z, \bar{z}) d z=-\iint_{D} \frac{\partial f}{\partial \bar{z}} d z d \bar{z}$$

where $\gamma$ is a contour in the complex plane enclosing the region $D$ and $f$ is sufficiently differentiable.