The precise form of singular functions, singular function representation and the extraction form for the stress intensity factor play an important role in the singular function methods to deal with the domain singularities for the Poisson problems with most common boundary conditions, e.q. Dirichlet or Mixed boundary condition [2, 4]. In this paper we give an elementary step to get the singular functions of the solution for Poisson problem with Neumann boundary condition or Robin boundary condition. We also give singular function representation and the extraction form for the stress intensity with a result showing the number of singular functions depending on the boundary conditions.