Abstract: In this paper the writer uses Muskhelishvili single-layer potential function solution and single crack solution for the torsion problem of a circular cylinder to discuss the torsion problem of a composite cylinder with an internal crack, and the problem is reduced to -a set of mixed-type integral equation with generalized Cauchy-kernel. Then, by using the integration formula of Gauss-Jacobi, the numerical method is established and several numerical examples are calculated. The torsional rigidity and the stress intensity factors are obtained. The results of these examples fit the results obtained by the previous papers better.