A muon bound to a nucleus can decay into an electron that is either free or remains in a bound state. We here consider the latter case, as part of a larger project of characterizing bound
muon decays. These studies are motivated by experimental searches for the lepton-flavor-violating $\mu \to e$ conversions in the field of a nucleus. Here we focus on the hypothetical case of close electron and muon masses. In this case we present a simple analytical formula for the branching ratio and compare it with a previous publication.