Abstract

The problem of asymptotic expansions of special functions with respect to indices or parameters arises in connection with the certain classes of index integrals and transformations with respect to the index, when in one of the formulas the integration is carried out over a parameter (index) of the kernel. The most common kernels of such transformations are hypergeometric functions, in particular, Bessel functions and their combinations. For such functions, it is true that the Mellin transformation has the ratio of the products of Euler's gamma functions, the asymptotics of which, in accordance with the Stirling formula, is known. The paper presents the Stirling formula for the gamma function of a complex argument, in which the imaginary part increases indefinitely, and the real part is fixed. It is found that for large values of the parameter, the asymptotic estimates of the modified Bessel functions of the imaginary index and their combinations contain the same multipliers of the independent argument, which lead to Fourier integrals. The asymptotics of Fourier integrals essentially depends on the differential properties of the integral function over the entire domain of integration. In this paper, the stationary phase method is used to study the asymptotics of Fourier integrals for large values of the parameter. According to the principle of localization, the contributions to the asymptotics of the integral from the critical points (stationary points of the phase) and from the boundary of the integration region are calculated.