Question

State and prove harnacks theorem

Answer 1

Answer:

Step-by-step explanation:

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 {\displaystyle u_{1}(z)} {\displaystyle u_{1}(z)}, {\displaystyle u_{2}(z)} {\displaystyle u_{2}(z)}, ... are harmonic in an open connected subset {\displaystyle G} G of the complex plane C, and

{\displaystyle u_{1}(z)\leq u_{2}(z)\leq ...} {\displaystyle u_{1}(z)\leq u_{2}(z)\leq ...}

in every point of {\displaystyle G} G, then the limit

{\displaystyle \lim _{n\to \infty }u_{n}(z)} {\displaystyle \lim _{n\to \infty }u_{n}(z)}

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

{\displaystyle u(z)=\lim _{n\to \infty }u_{n}(z)} {\displaystyle u(z)=\lim _{n\to \infty }u_{n}(z)}

is harmonic in the set {\displaystyle G} G.