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Because optical fibers that propagate one mode only are now the most likely candidate for future long-distance high-capacity communication, there is great interest in their light transmission characteristics. Standard descriptions of such fibers are rather unattractive both to students and to the physically minded scientist from several viewpoints, but mostly because numerical methods are always required at some stage in the analysis even if only to solve a transcendental eigenvalue equation. We reduce these restrictions by deriving simple analytical forms for the fields, propagation constant, and other parameters necessary to describe the fundamental mode on graded refractive index profiles. From these fully analytical expressions, we have a clearer understanding of propagation without recourse to numerical methods. Thus the theory of monomode fibers is now more accessible to undergraduates at elementary levels.