Functions of a Complex Variable with a Large Parameter and Construction of Regions

In this paper, a comparative analysis of the asymptotic estimate of integrals containing a large positive parameter is carried out. The need to study the asymptotic behavior of such integrals is explained by the fact that mathematical problems for various processes are reduced to the study of ordinary differential equations. Solutions of such equations are represented through special or other functions. Asymptotic behavior of such functions requires the study of integrals containing a large parameter. Two types of integral in the complex plane are considered. To obtain asymptotics, the saddle-point method is applied to one of the integrals, and the descent method is applied to the other. A detailed description is given in paper. Similarities and differences between these methods are revealed.