With the development of urban construction in China, more and more applications of rectangular tunnel are emerging, but there are few theoretical studies on rectangular tunnels. In this paper, the elastic theoretical calculation model of the semi-infinite space rectangular tunnel was established. The coefficients of conformal mapping function were determined by the least squares iterative method. And the calculated area was mapped to a concentric ring on the complex plane. Afterwards, The Muskhelishvili complex function method was used to expand the stress function in the calculation area into the form of Laurant series, which gives the zero stress boundary on the ground surface and the radial displacement boundary of the rectangular hole. The stress field and displacement field of the rectangular tunnel in the semi-infinite space under the given displacement condition were also obtained by the method. In this paper, the influence of aspect ratios, Poisson's ratios, and buried depths on the displacement field and stress field was analyzed, and the general rules of the displacement field and stress field of rectangular tunnels has been summarized. The results show that a smaller aspect ratio, a larger Poisson's ratio, and a smaller buried depth will make the settlement curve no longer similar to a Gaussian curve. The variation of these parameters will also affect the size and distribution of the stress field and displacement field to varying degrees.