Deriving the Cauchy-Riemann from the derivative of the complex conjugate.

Question

I am trying to prove the Cauchy-Riemann conditions naturally arise from the condition that:

$$\frac{\partial f(z,z^{*})}{\partial z^{*}} = 0$$

But I'm having trouble starting from the definition of a derivative and get stuck:

$$\frac{\partial f(z,z^{*})}{\partial z^{*}} = \lim_{\Delta z\to 0}\frac{f(z,z^{*}_{0}+\Delta z)-f(z,z^{*}_{0})}{\Delta z}$$

Am I going about this the right way? Is there a better way to do this?

Thanks!

Answer 1

HINT:

You've written the correct definition of the derivative. But instead, note that

$$ \frac{\partial f}{\partial z^*}=\frac12\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y} \right)\tag1 $$

Let $f=u+iv$ and set the right-hand of $(1)$ to $0$ . Can you finish now?