Аннотация:In present paper, we consider the problem of obtaining the asymptotics of solutions of a hyperbolic equation with holomorphic coefficients depending on the time parameter in the space of functions of exponential growth. Sufficient conditions for the convergence of asymptotic series presented in the asymptotics of solutions to a boundary value problem at infinity are also found. In the paper a new type of semiclassical asymptotics is obtained, which contains non-asymptotic, generally speaking, divergent series, requiring a summation method, and the resulting series converge in a certain neighborhood of zero. In the future, the approach presented in the article can be applied to the study of the semiclassical asymptotic behavior of the Laplace and heat equations at infinity. In the case where equations with meromorphic coefficients, as well as to second-order ordinary differential equations with irregular singularities.