This paper is a treatment of the three-dimensional nonaxisymmetric thermal-stress problem in a finite polygonal cylinder with a cylindrical circular hole under a steady-state temperature field. The problem is formulated in terms of the thermoelastic displacement potential and Boussinesq's, Dougall's, and Neuber's stress functions in cylindrical coordinates, which seem particularly convenient and have the advantage of being applicable to nonaxisymmetric problems. The problem is among the most difficult to solve by the usual analytical approach; therefore, the unknown coefficients involved in these functions are approximated by the boundary-point least-squares method.