A complete set of potential functions consisting of three scalar functions is presented for coupled displacement-temperature equations of motion and heat equation for an arbitrary x3-convex domain containing a linear thermoelastic transversely isotropic material, where the x3-axis is parallel to the axis of symmetry of the material. The proof of the completeness theorem is based on a retarded logarithmic potential function, retarded Newtonian potential function, repeated wave equation, the extended Boggio's theorem for the transversely isotropic axially convex domain, and the existence of a solution for the heat equation. It is shown that the solution degenerates to a set of complete potential functions for elastodynamics and elastostatics under certain conditions. In a special case, the number of potential functions is reduced to one, and the required conditions are discussed. Another special case involves the rotationally symmetric configuration with respect to the axis of symmetry of the material.