Proving that a Function is Analytic Given that it is Equal to the Complex Conjugate of an analytic Function
$$\lim_{h\to 0}\frac{\overline{f(\overline{z+h})}-\overline{f(\overline{z})}}{h}= \lim_{h\to 0}\frac{\overline{{f(\overline{z}+\overline{h})-f(\overline{z})}}}{h}=\overline{\lim_{h\to 0}\frac{{f(\overline{z}+\overline{h})-f(\overline{z})}}{\overline{h}}}=\overline{f'(\overline{z})} $$