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

TL;DR

This paper derives uniform upper bounds and asymptotic expansions for Macdonald's function, enabling analysis of the summability of divergent Kontorovich-Lebedev integrals for certain entire functions.

Contribution

It provides explicit bounds and asymptotic formulas for Macdonald's function, addressing the convergence of Kontorovich-Lebedev integrals for exponential type functions.

Findings

Established uniform upper bounds for Macdonald's function

Derived asymptotic expansion with explicit remainder term

Confirmed convergence of Kontorovich-Lebedev integrals for even entire functions

Abstract

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.