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

Active exterior cloaking for the 2D Helmholtz equation with complex wavenumbers and application to thermal cloaking