Embedding acoustic black holes (ABHs) on beams and plates has revealed as an appealing passive method for noise and vibration reduction. However, most ABH designs to date only concern straight beams and flat plates, while cylindrical structures are commonly found in the aeronautical and naval sectors. In this work, we suggest a semi-analytical method to compute the vibration field of a cylinder with an ABH indentation. We also show the ABH efficiency in terms of shell vibration reduction. It is proposed to resort to Gaussian basis functions in the framework of the Rayleigh-Ritz method, to reproduce the ABH cylinder vibration field. The ABH shell displacements in the three directions are decomposed in terms of Gaussian functions, which can be dilated and translated analogously to what is done with wavelet transforms. The functions are also forced to satisfy the continuity periodic conditions in the shell circumferential direction. The Gaussian expansion method (GEM) results in high precision at a low computational cost. The suggested semi-analytical method is validated against a detailed finite element (FEM) model. Modal frequencies and modal shapes are recovered very accurately. Besides, the mean square velocity of the annular ABH shell under point external excitation is compared to that of a uniform shell, in the 50-1000 Hz frequency range. Noticeable vibration reduction is achieved.