In this talk, we derive bounds on Stieltjes functions which generalize those provided by M. Gustafsson and D. D. Söjberg and apply them in the context of broadband cloaking to negatively answer the following challenging question. For the quasi-static approximation of Maxwell’s equation, is it possible to construct a passive cloaking device that will cloak an object over a whole frequency band? Common work with Graeme W. Milton