Analytic capacity of compact plane sets was first introduced by Ahlfors in order to study Painlev\'e's problem of finding a geometric characterization of the compact sets that are removable for bounded holomorphic functions. Despite recent advances due to Tolsa, the properties of analytic capacity remain rather mysterious. In particular, it is still unknown if analytic capacity is equal to the so-called Cauchy capacity. In this talk, I will present some new sufficient conditions for a compact planar set to have equal analytic and Cauchy capcities. As a consequence, I will describe how to produce examples of compact plane sets such that the above equality holds but the Ahlfors function is not the Cauchy transform of any complex Borel measure supported on the set.