# ${}$ Cauchy-Riemann equations

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?

## migrated from physics.stackexchange.comSep 27 at 15:20

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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: https://math.stackexchange.com/q/85648/