We calculate exactly the Coulomb electrostatic potential created by a uniformly charged square on its plane. The chosen solution method leads to certain types of integral expressions which involve combinations of products of error functions and power functions. These type of integrals that should be solved are not standard ones, therefore, the integration formulas we derive in the process may be deemed of mathematical interest on their own right. Use of these formulas enables us to derive an exact analytic expression for the Coulomb electrostatic potential created by a uniformly charged square at an arbitrary point on its plane. © 2012 Elsevier B.V.