  We design sources for the two-dimensional Helmholtz equation that can cloak
an object by cancelling out the incident field in a region, without the sources
completely surrounding the object to hide. As in previous work for real
positive wavenumbers, the sources are also determined by the Green identities.
The novelty is that we prove that the same approach works for complex
wavenumbers which makes it applicable to a variety of media, including media
with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's
addition formulas with complex arguments, we obtain new estimates that allow to
quantify the quality of the cloaking effect. We illustrate our results by
applying them to achieve active exterior cloaking for the heat equation.
