Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function $K_{ i\tau}(x)$. 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. [6]) whether these integrals converge for even entire functions of the exponential type in a weak sense.
Categories: math.CA