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C. Results from Analytic Function Theory

C.5 Derivatives and Differentials

Let $w=f(z)$ be a given complex function of the complex variable $z$. Then $w$ is said to have a derivative at $z_0$ if


\begin{displaymath}\lim_{\Delta z \rightarrow 0}\frac{f(z_0+\Delta z )-f(z_0)}{\Delta z } \end{displaymath} (C.5.1)

exists and is independent of the direction of $ \Delta z $. We denote this limit, when it exists, by $f'(z_0)$.