Complex differentiability

Definition

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be a function, and let $z_0 \in U$. The function $f$ is complex differentiable at $z_0$ if the limit $$f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$$ exists in $\mathbb{C}$. Here $h$ runs through the nonzero complex numbers with $z_0 + h \in U$. The number $f'(z_0)$ is the complex derivative of $f$ at $z_0$. The limit means: for each $\varepsilon > 0$ there is $\delta > 0$ such that $\left| \frac{f(z_0 + h) - f(z_0)}{h} - f'(z_0) \right| < \varepsilon$ for all such $h$ with $0 < |h| < \delta$.

Remarks

For example, $f(z) = z^2$ is complex differentiable at each $z_0$ with $f'(z_0) = 2 z_0$, because $\frac{(z_0 + h)^2 - z_0^2}{h} = 2 z_0 + h$. The number $h$ can go to $0$ from each direction in the plane. This makes the condition much stronger than real differentiability, see the conjugate is not holomorphic. If $f$ is complex differentiable at $z_0$, then $f$ is continuous at $z_0$. The Cauchy–Riemann equations describe complex differentiability with partial derivatives.

Explanations

L1 human

The definition is the same as the definition of the derivative of a real function, but $h$ is now a complex number. The quotient must have the same limit when $h$ goes to $0$ along the real axis, along the imaginary axis, or along any other path.