Harnack's principle

In complex analysis, Harnack's principle or Harnack's theorem is one of several closely related theorems about the convergence of sequences of harmonic functions, that follow from Harnack's inequality.

If the functions u 1 ( z ) , u 2 ( z ) , ... are harmonic in an open connected subset G of the complex plane C, and

in every point of G , then the limit

either is infinite in every point of the domain G or it is finite in every point of the domain, in both cases uniformly in each compact subset of G . In the latter case, the function

is harmonic in the set G .