re expressing the Cauchy Riemann Equations

Question

$$ \begin{split} \frac{\partial f}{\partial z} &= \frac{\partial f}{\partial x} \frac{\partial x}{\partial z} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial z}\\ &= \frac12 \left(\frac{\partial f}{\partial x} -i \frac{\partial f}{\partial y} \right) \end{split} $$

I saw the Cauchy-Riemann on wolfram alpha and am confused as to how the second equality comes about, any insight would be great!

http://mathworld.wolfram.com/Cauchy-RiemannEquations.html

Answer 1

To deduce the second equality it is sufficient to note that, since $z=x+iy$ (and $\bar{z}=x-iy$), then $$ x=\frac{1}{2}(z+\bar z)\quad y=-\frac{i}{2}(z-\bar z) $$ so $$ \frac{\partial x}{\partial z}=\frac{1}{2}\quad\frac{\partial y}{\partial z}=-\frac{i}{2} $$