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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.

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

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