Complex derivative

$f'(z_0)=\lim_{h\to 0} \dfrac{f(z_0+h)-f(z_0)}{h}$ , where $h$ is a complex number approaching $0$ from any direction.

The complex derivative of $f$ at $z_0$ is $f'(z_0) = \lim_{h\to 0} \dfrac{f(z_0+h)-f(z_0)}{h}$ , where $h$ is itself a complex number and the limit must come out the same regardless of the direction $h$ approaches $0$ from. Requiring agreement across every direction of approach -- not just along the real axis, as in single-variable calculus -- is what makes complex differentiability such a demanding condition, and why it forces far more structure on $f$ than real differentiability ever does.

Examples

For $f(z)=z^2$ , $f'(z_0)=\lim_{h\to 0}\dfrac{(z_0+h)^2-z_0^2}{h} = \lim_{h\to 0}(2z_0+h) = 2z_0$ .
$f(z)=\bar z$ (complex conjugation) has no complex derivative anywhere, since the difference quotient gives a different limiting value depending on whether $h$ approaches $0$ along the real or imaginary axis.