A new transform pair representing solutions to the complex Helmholtz equation in a convex twodimensional polygon is derived using the theory of Bessel’s functions and Green’s second identity. The
derivation is a direct extension of that given by Crowdy [IMA J. Appl. Math, 80, (2015)] for “FourierMellin transform” pairs associated with Laplace’s equation in various domain geometries. It is shown
how the new transform pair fits into the collection of ideas known as the Fokas transform where the key
step in solving any given boundary value problem is the analysis of a global relation. Here we contextualize those global relations from the point of view of “reciprocal theorems” which are familiar tools in
the study of the effective properties of physical systems. A survey of the many uses of this new transform approach to the complex Helmholtz equation in applications is given. This includes calculation
of effective impedance in electrochemical impedance spectroscopy and in other spectroscopy methods
in diffusion wave field theory, application to the 3w method for measuring thermal conductivity and to
unsteady Stokes flow. A theoretical connection between this analysis of the global relations and Lorentz
reciprocity in mathematical physics is also pointed out.