  The sparse approximation of high-frequency Helmholtz-type integral operators
has many important physical applications such as problems in wave propagation
and wave scattering. The discrete system matrices are huge and densely
populated; hence their sparse approximation is of outstanding importance. In
our paper we will generalize the directional $\mathcal{H}^{2}$-matrix
techniques from the \textquotedblleft pure\textquotedblright\ Helmholtz
operator $\mathcal{L}u=-\Delta u+\zeta^{2}u$ with $\zeta=-\operatorname*{i}k$,
$k\in\mathbb{R}$, to general complex frequencies $\zeta\in\mathbb{C}$ with
$\operatorname{Re}\zeta>0$. In this case, the fundamental solution decreases
exponentially for large arguments. We will develop a new admissibility
condition which contains $\operatorname{Re}\zeta$ in an explicit way and
introduce the approximation of the integral kernel function on admissible
blocks in terms of frequency-dependent \textit{directional expansion
functions}. We develop an error analysis which is explicit with respect to the
expansion order and with respect to $\operatorname{Re}\zeta$ and
$\operatorname{Im}\zeta$. This allows to choose the \textit{variable }expansion
order in a quasi-optimal way depending on $\operatorname{Re}\zeta$ but
independent of, possibly large, $\operatorname{Im}\zeta$. The complexity
analysis is explicit with respect to $\operatorname{Re}\zeta$ and
$\operatorname{Im}\zeta$ and shows how higher values of $\operatorname{Re}%
\zeta$ reduce the complexity. In certain cases, it even turns out that the
discrete matrix can be replaced by its nearfield part.
  Numerical experiments illustrate the sharpness of the derived estimates and
the efficiency of our sparse approximation.
