I am wondering this because I just came from my PDE's 2 class and we talked about the regularity of the Laplace equation and elliptic functions DE's. My professor the stated the following which we didn't prove:
If $\bigtriangleup u = 0$ then $u$ is analytic $\Rightarrow$ we can express $u$ as a Taylor Series
I found this very interesting and I was wondering if the same fact would hold for the Poisson Equation ($\bigtriangleup u=f$ ) does this give that $u$ must also be analytic ? If so I would love to see a proof as I haven't been able to find a proof showing it is or is not true.