The stress field in the cylindrical shell with the circular hole is considered. Thoroughly analyzing previous works we concluded that up to the present there are no explicit formulae useful for applications despite numerous works on this subject. Moreover, the classical analytical approach lacks a rigorous mathematical foundation. Its applicability is limited to cases that slightly differ from the plane Kirsch problem. The numerical results of various researches show a significant discrepancy. This paper proposes a new analytical approach based on a different form of representation for the fundamental system of solutions of governing equations. Since, in contrast to previous works, cutting of series is not required the boundary conditions are satisfied with a very high degree of accuracy. Our model works in all ranges that mechanics allows and there are no mathematical restrictions. Some numerical results are presented.