Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the entire complex plane. So, just by knowing how the function behaves at a teenie-weenie open disc in $\mathbb{C}$, the behavior at a much larger scale is uniquely defined.
Does this extraordinarily beautiful property have any direct applications in physics? For example, you know how some kind of field looks in a tiny area and you can somehow assume it is described by a holomorphic function and by analytic continuation you "extrapolate" the field to a much bigger domain.
EDIT: I cross-posted this on physics.SE, it might fit better there.