Cauchy-Riemann equations - a doubt about the proof on MathWorld

Question

This may be a stupid question but I was reading the proof of Cauchy-Riemann equations from Wolfram MathWorld. Here, they seem to have put $\frac{\partial x}{\partial z} = \frac{1}{2}$. Shouldn't this be 1 as $z=x+iy$ and $x$ and $y$ are independent variables?

Answer 1

This should be migrated to Math.

$x$ can be written as a function of $z$ as $x = \frac{z+z^*}{2}$

Differentiating with respect to $z$ we get $\frac{\partial x}{\partial z} = \frac{1}{2}$

One part that may seem odd is that the complex conjugate of $z$ is treated as a constant under the differentiation with respect to $z$. An in-depth answer to why this is can be found here: Why can the complex conjugate of a variable be treated as a constant when differentiating with respect to that variable?