# Is the solution to the Poisson equation an analytic function in general?

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.

• Well, if $f$ is non-analytic, and $\Delta u=f$, then $u$ cannot be analytic... Feb 13, 2019 at 17:07