In this paper new exact Green's functions and new exact Poisson-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a half-wedge with mixed homogeneous mechanical boundary conditions (the boundary angle is free of loadings and normal displacements and tangential stresses are prescribed on the boundary quarter-planes) are derived. The thermoelastic displacements are produced by a heat source applied in the inner points of the half-wedge and by mixed non-homogeneous boundary heat conditions (the temperature is prescribed on the boundary angle and the heat fluxes are given on the boundary quarter-planes). When thermoelastic Green's function is derived the thermoelastic displacements are generated by an inner unit point heat source, described by -Dirac's function. All results are obtained in terms of elementary functions and they are formulated in a special theorem. Analogous results for an octant and for a quarter-space as particular cases of the angle of the thermoelastic half-wedge also are obtained. The main difficulties to obtain these results are in deriving the functions of influence of a unit concentrated force onto elastic volume dilatation (q) and, also, in calculating a volume integral of the product of function (q) and Green's function in heat conduction. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for a quarter-space and a half-wedge, using the derived Poisson-type integral formula and the influence functions (q) also are included. The proposed approach may be extended not only for many different BVPs for half-wedge, but also for many canonical cylindrical and other orthogonal domains.