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Uniform upper bounds and the asymptotic expansion with an explicit remainderterm are established for the Macdonald function $K_{i\tau}(x)$. The results canbe applied, for instance, to study the summability of the divergentKontorovich-Lebedev integrals in the sense of Jones. Namely, we answeraffirmatively a question (cf. [6]) whether these integrals converge for evenentire functions of the exponential type in a weak sense.

Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals