Vanishing of harmonic functions in planar domains - Wang,Ruolin (Bachelor's Theses in Mathematical Sciences, 2024)
We investigate the vanishing conditions for harmonic functions in planar domains. We first prove that if a complex-valued harmonic function $u$ in an open set $Ω \subseteq \mathbb{C}$ vanishes at a point $a \in Ω \cap \mathbb{R}$ and satisfies $(\partial+\bar{\partial})^k u(a)=0$ for all $k ≥ 0$, then $u$ vanishes identically on the connected component of $Ω \cap \mathbb{R}$ containing a, where $ \partial=\frac{1}{2}(\frac{\partial}{\partial x} +\frac{1}{i}\frac{\partial}{\partial y})\quad \text{and}\quad \bar \partial=\frac{1}{2}(\frac{\partial}{\partial x} -\frac{1}{i}\frac{\partial}{\partial y}). $ Additionally, we show that if $u$ vanishes on a circular arc within its domain, then $u \equiv 0$ in $Ω$. These results extend classical uniqueness theorems for harmonic functions and are validated through analysis in both the open unit disc $D$ and general planar domains.
Keywords: Harmonic functions; uniqueness theorem; power series expansion