Let $R,\phi$ be real valued $C^1$ functions over a complex variable $z$.
Consider the following function: $f=Re^{i\phi}$ .
Where we define $f_{\overline{z}}= \frac{1}{2}(f_x+if_y)$ I want to differentiate this function with "respect to $\overline{z}$". I proved that this rule of derivation also has the property on the multiplication. $(fg)_\overline{z}=fg_{\overline{z}}+gf_{\overline{z}}$ So:
$f_{\overline{z}}=R_\overline{z}e^{i\phi}+R(e^{i\phi})_{\overline{z}}$ But I don't know how to differentiate: $(e^{i\phi})_{\overline{z}}=\frac{1}{2}((e^{i\phi(z)})_{{x}}+(e^{i\phi(z)})_{{y}})$ I don't know what can I do in this step, Even if I view the function as a function over two real variables $(x,y)=z$ :/ Please help me