A generalized higher−order WKB approximation is found for the set of equations h″j(t) + u2 𝒥Nk=1 Mjk(t)hk(t) = 0 (u → ∞), when the coefficients Mjk form a positive definite Hermitian matrix M satisfying a smoothness condition as a function of t. In the construction, essential use is made of a transformation introduced by Kato to connect smoothly the eigenvectors of M(t) at different values of t. Eigenvalue degeneracies which exist for all t are covered by the method. The expansion breaks down at points t where the multiplicities of the eigenvalues of M(t) change; this phenomenon, analogous to the ’’turning point’’ problem of the ordinary WKB method, will be studied in a second paper. The asymptotic nature of the expansion is proved; error bounds can be extracted from the proof but are not studied here.