Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function Kiτ(x)$K_{i\tau }(x)$$K_{i\tau }(x)$. The results can be applied, for instance, to study the summability of the divergent Kontorovich-Lebedev integrals in the sense of Jones. Namely, we answer affirmatively a question (cf. [Ehrenmark U. Summability experiments with a class of divergent inverse Kontorovich-Lebedev transforms. Comput Math Appl. 2018;76(1):141–154.]) whether these integrals converge for even entire functions of the exponential type in a weak sense.

为麦克唐纳函数建立了一致上限和具有显式余项的渐近展开式Kτ _( x )$K_{我\tau }（X）$$K_{i\tau }(x)$。例如，结果可用于研究琼斯意义上的发散 Kontorovich-Lebedev 积分的可求和性。也就是说，我们肯定地回答了一个问题（参见 [Ehrenmark U. Summability Experiments with a Class of Divergent Inverse Kontorovich-Lebedev Transforms.Comput Math Appl. 2018;76(1):141–154.]）这些积分是否收敛于偶数弱意义上的指数型的全部函数。