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The evanescent wave theory of modal propagation in graded index optical fiber developed recently by Choudhary and Felsen is based on certain postulates, which are examined here. It is shown that an analytical condition on the asymptotic expansion coefficents for the modal amplitudes on the fiber axis, as imposed by these authors, should be replaced by the single-valuedness of these coefficients in a strip of the complex coordinate plane. The analyticity condition is of questionable validity because of the nonunifomity of the asymptotic expansion on the axis. The previous results of Choudhary and Felsen are found to be unaffected by this change but the method is now made rigorous and need not be justified, as before, by comparison with asymptotic expansions of exact solutions for special profiles. Also developed here is a uniform asymptotic approximation that is valid near the fiber axis and connects with the leading term of the nonuniform evanescent wave theory formulation. Within this rigorous framework, the evanescent wave theory continues to provide a very useful and systematic procedure for calculating modal eigenvaluess and modal fields to arbitrary orders in inverse powers of the large wavenumber k.