  We consider the Fokas method expression for the solution of the heat equation
on the half line with Dirichlet data and we study in detail its boundary
behaviour near the spatiotemporal domain boundaries, i.e., the semi-axes,
infinity and the origin, by analyzing the integrals involved. We also study the
boundary behaviour of the derivatives of the solution. In particular we give
conditions on the data which guarantee the extension of the solution to a
smooth function up to the semi-infinite boundaries.
