The hybrid method is employed in the numerical simulation of electromagnetic waves propagating in an axisymmetrical and longitudinally stratified medium. Two categories of basis functions are introduced, one of which sustains the amplitude and the other of which sustains the slope, of the function in question at nodal points. An interval extending to infinity is devised, whereby infinity is treated more reasonably than in previous works. The application of the above two contrivances makes possible the reduction of nodal points from 45, as necessitated by triangular basis functions, to 4 and also a two-orders-of-magnitude improvement of the overall computational efficiency. Such a substantial improvement will facilitate the inversion of induction logging. ¿ 1999 American Geophysical Union