  We provide a new approach to studying the Dirichlet-Neumann map for Laplace's
equation on a convex polygon using Fokas' unified method for boundary value
problems. By exploiting the complex analytic structure inherent in the unified
method, we provide new proofs of classical results using mainly complex
analytic techniques. The analysis takes place in a Banach space of complex
valued, analytic functions and the methodology is based on classical results
from complex analysis. Our approach gives way to new numerical treatments of
the underlying boundary value problem and the associated Dirichlet-Neumann map.
Using these new results we provide a family of well-posed weak problems
associated with the Dirichlet-Neumann map, and prove relevant coercivity
estimates so that standard techniques can be applied.
