Summary: A transversely isotropic linear elastic half-space with depth wise axis of material symmetry containing a cylindrical cavity of finite length is considered to be under the effect of a time-harmonic torsion force applied on a ring at an arbitrary depth on the surface of the cylindrical cavity. With the aid of cosine transforms, the boundary value problem for the fundamental solution is reduced to a generalized Cauchy singular integral equation. The Cauchy integral equation involved in this paper is analytically investigated and the final equation is numerically solved with an in-depth attention. Integral representation of the stress and displacement are obtained, and is shown that their degenerated form to the static problem of isotropic material is coincide with existing solutions in the literature. To investigate the effect of material anisotropy, the results are numerically evaluated and illustrated.