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I have a problem in a demonstration where I have to do a derivative $\frac{df}{dc}$ of

$f=(y-cx)(y-cx)^{*}$

where * denotes the complex conjugate.

In my demonstration the derivative $\frac{df}{dc}$is equal to 0. So I tried to solve it by applying the chain rule:

$x(y-cx)^{*}+x^{*}(y-cx)=0$

obtaining the result:

$c=\frac{xy^{*}+yx^{*}}{2xx^{*}}$

Unfortunately I should have obtained

$c=\frac{yx^{*}}{xx^{*}}$

Can you help me with this derivative of complex number? What am I doing wrongly?

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    $\begingroup$ $f=|y-cx|^2$ is not differentiable $\endgroup$
    – Raffaele
    Dec 2, 2020 at 18:35

1 Answer 1

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I'm confused; $c \mapsto \overline{c}$ is not analytic and $f=|y|+|cx|-\overline c \overline x y - cx\overline y$, so why should $f$ be complex-differentiable?

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