Diffusion through a half space: Equivalence between different formulations of the unique solution

Diffusion through a half space involves a classical parabolic partial differential equation that is encountered in many fields of physics and has significant engineering applications, concerning particularly heat and mass transfer. However, in the specialized literature, the solution is usually achieved restricting the problem to particular cases and attaining apparently different formulations, thus a comprehensive overview is hindered. In this paper, the solution of the diffusion equation in a half space with a boundary condition of the first kind is worked out by means of the Fourier's Transform, the Green's function and the similarity variable, with a proof of equivalence - not found elsewhere - of these different approaches. The keystone of the proof rests on the square completion method applied to Gaussian-like integrals, widely used in Quantum Field Theory.