| ¡@¡@The Helmholtz eigenvalue problems in two dimensions is classical in mathematics and physics. Nevertheless, analytical methods for estimating the eigenvalues are still of much current interest. ¡@¡@In this talk, the region concerned is a bounded doubly connected region with the inner boundary which encloses a region of maximal dimension 2c , c " 1. A modified perturbation method is formulated by applying perturbation method, reflection method, and Fredholm alternative theorem. This method provides a general formula for the asymptotic expansion ( c ¡÷ 0 ) of the lowest eigenvalue. The first three order terms of the asymptotic expansion are computed explicitly by correcting the inner and the outer boundary conditions alternately and by applying the generalized Green's functions. The resulting approximations are compared with the exact solutions and with the approximations determined by other investigators. The relations between the first three order terms of the asymptotic expansion and geometric properties of the region are also investigated. |