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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

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You differentiate a function to find its derivative. It's confusing, I know. – Rahul Aug 24 '12 at 23:41
Now I think "derivate" should be made a word :) – Tunococ Aug 25 '12 at 1:42
up vote 1 down vote accepted

Hint - try using: $$e^{i\phi} = \cos(\phi) + i\sin(\phi)$$

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