A general analytic formula is obtained for the two-center Coulomb integrals over Slater-type orbitals in elliptical coordinates. Finite series expansions are used in the evaluation of the radial part of the integrals. The analytic formula is expressed in terms of a product of the well-known auxiliary functions Ak(p) and Bk(p) and incomplete gamma functions. Recursive relations for the computer evaluation of these functions are given as well. The recursive relations are stable and our computer results are in good agreement with the benchmark values given in the literature.