The Maximum Principle

Abstract

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.