I've got a problem to determine if $z \rightarrow \cos(\bar{z})$ is differentiable on $\mathbb{C}$. Actually, I tried to determine if :
$\lim \frac{\cos(\bar{z+h})-\cos(\bar{z})}{h}$ exists when $h \rightarrow 0$, and to do that, I tried to use the formula for $\cos(z)$ (ie $\cos(z) = \frac{e^{iz}+e^{-iz}}{2}$, but that doesn't help me...
Someone could help me, please? Thank you.
No, it is not differentiable. If $z=x+yi$, with $x,y\in\mathbb R$, then\begin{align}\cos(\overline z)&=\cos(x-yi)\\&=\cos(x)\cos(yi)+\sin(x)\sin(yi)\\&=\cos(x)\cosh(y)+\sin(x)\sinh(y)i.\end{align}Now, use the Cauchy-Riemann equations.