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We consider the zeros on the boundary $\partial \Omega$ of a Neumanneigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. Weprove that the number of its boundary zeros is $O (\lambda)$ where $-\Delta\phi_{\lambda} = \lambda^2 \phi_{\lambda}$. We also prove that the number ofboundary critical points of either a Neumann or Dirichlet eigenfunction is$O(\lambda)$. It follows that the number of nodal lines of $\phi_{\lambda}$(components of the nodal set) which touch the boundary is of order $\lambda$.This upper bound is of the same order of magnitude as the length of the totalnodal line, but is the square root of the Courant bound on the number of nodalcomponents in the interior. More generally, the results are proved forpiecewise analytic domains.

Counting Nodal Lines Which Touch the Boundary of an Analytic Domain