= Complex differentiability
{title2=$f'(z)=\lim_{h\to0}(f(z+h)-f(z))/h$}
A complex-valued <function> is complex differentiable at $z_0$ if
$$
f'(z_0)=\lim_{\substack{h\to0\\h\in\mathbb C\setminus\{0\}}}\frac{f(z_0+h)-f(z_0)}{h}
$$
exists. The same limit must hold for every approach, including curved paths. Equivalently,
$$
f(z_0+h)=f(z_0)+f'(z_0)h+o(|h|).
$$
The error divided by $|h|$ tends to zero. Thus the <derivative> is a single complex multiplier, not an arbitrary real <linear map> of the plane. If the real coordinate map is differentiable, <complex differentiability> is equivalent to its satisfying the <Cauchy-Riemann equations> at that point.
The simplest failure is <complex conjugation>, $f(z)=\overline z$. At every $z_0$ its increment quotient is $\overline h/h$. For real $h=t\ne0$ it is $1$, while for $h=it$ it is $-1$. More generally, $h=te^{i\theta}$ gives $e^{-2i\theta}$. Although <complex conjugation> is a smooth real reflection and has every real <directional derivative>, these incompatible quotients rule out a complex <derivative> everywhere.
Agreement on two axes is necessary, but does not by itself establish a limit over the whole <complex plane>. The <Cauchy-Riemann equations> need the real differentiability condition above, or a sufficient regularity condition such as continuous <partial derivatives> near the point. There is a second distinction: $f(z)=|z|^2$ has complex <derivative> zero at zero because its quotient there is $\overline h\to0$, but its <Cauchy-Riemann equations> fail at every nonzero point. It is complex differentiable at zero without being a <holomorphic function> on any neighbourhood of zero.
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