Asymptotic behavior of a function of a complex variable with a parameter and zeros

The study of the asymptotic behavior of solutions of singularly perturbed equations in complex domains reduces to the study of integrals of exponential functions of a complex variable with a parameter. The problem is to study the asymptotic behavior of such functions. The functions in the exponent have zeros. The study of such functions is hampered by the fact that it is necessary to select a certain part from a given region and choose integration paths that ensure the boundedness of the considered functions with respect to a small parameter. The saddle point method is not applicable to such integrals. To solve the problem, the level lines of harmonic functions generated by analytic functions are applied. The level lines divide the area in the complex plane into parts. Integration paths are chosen that ensure boundedness of integrals with respect to a small parameter. Boundary-layer lines, areas where the integrals have no limit with respect to a small parameter, but are limited with respect to the modulus, are revealed; regular domains (integrals have a limit); singular regions (integrals are not limited). All constructions are accompanied by corresponding drawings. In the future, the results of this paper can be used for the theory of singularly perturbed equations in a complex domain.