Anti-plane problems of elasticity theory occupy an important place in the mechanics of deformable solids, which is connected with their role in modeling of various engineering problems. This work is devoted to solving of the anti-plane problem of the elasticity theory for a strip weakened by a crack, the construction of an analytical solution of the problem through the application of the integral Fourier transform, the reduction of the original problem to one-dimensional problem and by its solving through the Green’s function. The final results are derived by the solving of the singular integral equation by the orthogonal polynomials method. The numerical solution of the problem consists in the application of reduction method and deriving the unknown jump function in the form of a series of Chebyshev polynomials. The study of displacements and stresses is made through the construction of graphic calculations for the area and the calculation of stress intensity factors for this problem under specified loads and for specific parameters of materials and crack locations.