Let $K\subset A\subset C$, where $K$ is compact, $A$ is open, and $C$ is the complex numbers. Let $f:C\rightarrow C$ be holomorphic on $A$ (holomorphic = complex derivative exists for all points in $A$). Is it true that on $K$, $[f(z+\Delta z) - f(z)]/\Delta z$ converges uniformly as $\Delta z\rightarrow 0$?
2026-04-23 06:54:51.1776927291
For a holomorphic function, does Newton's difference quotient converge uniformly on compact sets?
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