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Sep 30, 2021 at 23:17 comment added Tom Copeland Also harmonic functions, i.e., conjugate solns. to Laplace's equation, and their associated mean value and max/min principles can be used to give equivalent definitions of analtytic complex functions. This is closely related to electrostatics and to Brownian motion, so physics and probabilistic approaches are possible. Other approaches not mentioned already?
Sep 27, 2021 at 16:03 comment added Timothy Chow This example was also suggested by Sam Hopkins.
S Sep 27, 2021 at 16:00 history answered Michael CC BY-SA 4.0
S Sep 27, 2021 at 16:00 history made wiki Post Made Community Wiki by Michael