  A new transform pair which can be used to solve mixed boundary value problems
for Laplace's equation and the complex Helmholtz equation in bounded convex
planar domains is presented. This work is an extension of Crowdy (2015, CMFT,
15, 655--687) where new transform techniques were developed for boundary value
problems for Laplace's equation in circular domains. The key ingredient of the
method is the analysis of the so called global relation which provides a
coupling of integral transforms of the given boundary data and of the unknown
boundary values. Three problems which involve mixed boundary conditions are
solved in detail, as well as numerically implemented, to illustrate how to
apply the new approach.
