 Polygonal modified Helmholtz global relation

For $u_{z\bar{z}}={\beta }^{2}u$, pull back its spectral closed differential one-form along each straight side $z=m_j+s{h}_j$. Under counterclockwise traversal and outward normal derivatives $q_j$, the side density is $i{e}^{-i\beta (\lambda z-\bar{z}/\lambda )}[|h_j|q_j+\beta (\lambda h_j+{\bar{h}}_j/\lambda )g_j]$, where $g_j$ is the Dirichlet boundary data. The Generalized Stokes theorem gives the displayed global relation. Clockwise traversal reverses the normal-derivative coefficient; all orientations must be changed consistently.