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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 $\Omega \subseteq \mathbb{C}$ vanishes at a point $a \in \Omega \cap \mathbb{R}$ and satisfies $(\partial+\bar{\partial})^k u(a)=0$ for all $k \geq 0$, then $u$ vanishes identically on the connected component of $\Omega \cap \mathbb{R}$ cont
