It is known that in lossless and closed guides filled with gyroelectric or gyromagnetic media, Maxwell's equations are transformed into an infinite linear algebraic equation system by application of the Galerkin version of Moment method. Propagation constant of the problem whose exact solution is not known is found as the square root of the eigenvalue of the coefficient matrix of this infinite linear algebraic equation system. However the quantities determined from the Moment method do not have much physical meaning by themselves. This is why it is desired to express the propagation constant as a function of frequency. Therefore with the help of algebraic function theory we attempt to express eigenvalues of the coefficient matrix, i.e., the squares of the propagation constants, as a series expansion. By this technique Laurent and Puiseux series expansions in the neighborhood of singular points are obtained for these functions. The series expansions which are obtained by the algebraic function approximation in the eigenvalue problems of closed, lossless, uniform waveguides, bring about a function theoretic insight in order to investigate the properties of the propagation constant functions. In previous work reported in the literature there exists a derivation of these expansions accomplished by using differentiation of the characteristic equation of above coefficient matrix, considering it as an implicit function of the propagation constant function and the frequency. In this work computation of the necessary expansion coefficients of the Laurent and Puiseux series, is achieved by the least squares technique (LST) which is a curve fitting method. In this way we attempt to find a simple solution to the problem of computation of these coefficients, which dates back to Sir Isaac Newton and in general which is rather tedious.