A Lipschitz-Hankel integral of \(K_ 0\) (Q1121412)

The author considers an incomplete Lipschitz-Hankel integral \[ (1)\quad K_{e_ 0}(a,z)=\int^{z}_{0}e^{at} K_ 0(t)dt, \] where \(K_ 0(t)\) is the Bessel function of the imaginary argument (the MacDonald function). In an another paper of the author (to be published) it is shown, that (1) can be represented in closed-form in terms of elementary, MacDonald and Kampé de Fériet double hypergeometric functions. In this paper another representation for the integral (1) and related integrals \[ \int^{z}_{0}\cos at K_ 0(t)dt,\quad \int^{z}_{0}\sin at K_ 0(t)dt \] are given.