The Dunham coefficients are an indispensable part of the analysis of the ro-vibrational spectrum of a diatomic molecule. They provide a direct connection between the ro-vibrational states observed and the interatomic potential that must exist in the molecule. One may deduce the interatomic potential from the spectrum or, alternatively, predict the spectrum from a theoretically generated interatomic potential. The coefficients result from a mathematical analysis of the Schrodinger equation for such a system. Dunham's derivation relied on the WKB approximation and, as such, was subject to the associated limitations. In this paper, the quantum condition is derived without any reliance on the WKB approximation, using only principles of complex analysis. This sidesteps the need for introducing an approximate function to join solutions and suggests that the expansion has a more fundamental basis. Also in this paper, a mathematical algorithm for generating the Dunham coefficients is elaborated in detail. Careful attention is paid to keeping these coefficients accurate to a specified order in the smallness parameter, tau(e) = B-e/omega(e). This algorithm is intended to be presented in a way that makes it amenable to incorporation into computer code and points where the efficiency can be improved are indicated. Coefficients have been generated and presented in the literature through the years. The current paper presents the coefficients explicitly to tenth order in the smallness parameter, far more than have been generated in any previous work.