Let's $$f:\mathbb{C}\to\mathbb{R}$$ is holomorphic. In all lectures notes that I've read, said that $$f'(z)=\lim_{h\to 0}\frac{f(z+h) - f(z)}{h}.$$ Is it true. Or correctly next definition $$f'(z)=\lim_{h\to 0}\frac{f(z+h) - f(z)}{|h|}.$$ On the other hands, is it true that $f'(z)$ is complex?
Thanks