So we got two harmonic functions $u(x,y)$ and $v(x,y)$ on $D$. They agree in the neighborhood of some point $(x_0,y_0)$ $\in D$. Prove that they agree on whole set D.
how can I get from harmonic functions to analytic, in order to use Identity theorem? I don't know u and v are conjugate or not. I believe there was some theorem which stated that in open connected set function u has harmonic conjugate, but how can i be sure that this conjugate is function v?