  Formulae for the value of a harmonic function at the center of a rectangle
are found that involve boundary integrals. The central value of a harmonic
function is shown to be well approximated by the mean value of the function on
the boundary plus a very small number (often just 1 or 2) of additional
boundary integrals. The formulae are consequences of Steklov (spectral)
representations of the functions that converge exponentially at the center.
Similar approximation are found for the central values of solutions of Robin
and Neumann boundary value problems. The results are based on explicit
expressions for the Steklov eigenvalues and eigenfunctions.
