maksimumsprinsipp

The maximum principle, also known as the maximum modulus principle or the maximum value principle, is a fundamental theorem in complex analysis. It states that if a function is analytic (holomorphic) in a domain D and continuous on its boundary, then the maximum value of the absolute value of the function must occur on the boundary of the domain. In other words, the function cannot attain a maximum modulus inside the domain unless it is a constant function.

This principle has important implications. For a non-constant analytic function, the maximum value of its absolute value will be on the boundary. Conversely, the minimum value of the absolute value of a non-constant analytic function, if it is never zero in the domain, also occurs on the boundary. If the analytic function is zero within the domain, then the minimum modulus can occur inside.

The maximum principle is often applied to prove other theorems in complex analysis. It can be used to establish properties of analytic functions related to their growth and behavior within regions. For example, it plays a role in proving Schwarz's lemma, which provides bounds for analytic functions mapping the unit disk to itself. The principle is a direct consequence of Cauchy's integral formula and the properties of analytic functions, particularly their harmonic nature (the real and imaginary parts of an analytic function are harmonic functions).