Let $u(x,y)$ be harmonic on a domain $D \in \mathbb C$, Can we say that $f(x)=u(x,0)$ is real analytic on $D \cap \mathbb R$ if it is non-empty?
Clearly $f$ is infinitely differentiable on the domain, But what about analyticity?
2026-03-25 22:23:41.1774477421
Real analyticity of harmonic functions
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Yes, that's true. One way to see this is to show that there exists (locally) a conjugate harmonic function $v$ such that $u+iv$ is holomorphic. This is shown in text books on harmonic maps or elliptic PDE of second order.
Another way is to find an integral representation for $u$ (again, it suffices to do that locally. Googling will find you resources for this, e.g. this one) and to note that the integral kernel admits a represention by it's Taylor series.