The Maximum Principle

The values that an analytic function assumes in the different parts of its domain of existence are related to each other: they are connected by analytic continuation and it is impossible to modify the values in one part without inducing a change throughout. Therefore an analytic function can be compared to an organism the main characteristic of which is exactly this: Action on any part calls forth a reaction of the entire system. E.g. the propagation of convergence [251–258] can be compared to the spreading of an infection. Mr. Borel advanced ingenious reflections upon similar comparisons1. We shall examine in what manner the moduli of the values are related that the function assumes in different parts of its domain of existence.