Given,

u_xx + u_yy = 0

We know by the Cauchy-Riemann equations that if f(z) is analytic on D then Re{f(x)} and Im{f(z)} are solutions to the Laplace equation on D.

Is the converse true? And furthermore, if the converse is true can we give an explicit formula to solve Laplace equation satisfing the Dirichlet problem?