How can I find infinitely many functions that solve the Laplace equation? I think I need to find analytic functions in open subset , but I don't know how to use the Cauchy Riemann equations here to help me. Someone can help please.
Remember that every analytic function (in the open set ) can be seen as where are such that and satisfy and (with and ). Now assume that (if you haven't proved that the real and imaginary part of an anlytic function are of class) and using the C-R equations we get:
And since we have that the mixed partial derivatives of are equal and so, adding the last two equations together we get that satisfies the Laplace equation in (the same argument tells you that also satisfies Laplace in ). And so the real and imaginary part of an analytic function are solutions to the Laplace equation (ie. they're harmonic functions)