Abstract: The analytical stress solution for an infinite plate containing a single hole of arbitrary shape can be obtained by complex variable method. As to the doubly-connected domain problem that an infinite plate contains two round holes or two elliptical holes, it can also be solved using a variety of methods, such as the bi-polar coordinate method, the stress function method, the complex variable method, and the Schwarz alternating method. The complex variable method combined with conformal mapping is one of importance methods which can be used to obtain analytical stress solution, but it is not yet used to solve the problem of an infinite plate containing a square hole and an elliptical hole. Taking advantage of the conformal mapping method, the problem that an infinite plate contains two specific holes, which far-field uniform stress is applied at infinity and the boundaries of the two holes are subjected to uniform vertical compression, can be solved. The key step of this method is to find the corresponding mapping function with which the considered region can be mapped onto a ring in the image plane. Based on the Riemann mapping theorem, we propose a general form of the mapping function and figure out the concrete mapping function for the specific problem using optimization method. The basic equation set for solving the two analytical functions is established through the stress boundary condition of the two holes. Then the analytical stress solution can be obtained according to the two analytical functions. The analytical stress solution is compared with numerical stress solution of ANSYS finite element method. Effects of separation distance, size of elliptical hole, and the orientation of holes on tangential stress of the boundary is investigated using the newly derived solution. The stress distributions on the line that connects the centers of the two holes under different loads are presented.