Let \(U\subset \mathbb{C}\) be an open domain on the complex plane. We call a function \(f\colon U \to \mathbb{C}\) (complex) differentiable at \(z_0\in U\) if
\begin{equation*} f'(z_0)=\lim_{z \to z_0}\frac{f(z)-f(z_0)}{z-z_0} \end{equation*}exists. This value is the derivative of \(f\) at \(z_0\).
If a function is differentiable at every point we call it holomorphic and denote the derivative of \(f\) as \(f'\).
Remarks
- There are alternative ways of defining differentiability.
- A holomorphic function is continuous.
- The conjugation is not holomorphic.
- A function is holomorphic if and only if it satisfies the Cauchy-Riemann equations .