It is the essential prerequisite for mechanical analysis of the underground cavern with the complex variable theory that the conformal mapping function must be solved. According to the boundary correspondence principle，a conformal mapping function is established that can transform the exterior of the arbitrary excavation cross-section in half-plane to the area between two concentric circles. A method for calculating the conformal mapping coefficients is also proposed. By using the method of odd even point interpolation and genetic optimization and sequential quadratic programming algorithm，the high resolution is realized. Based on the analytic property of mapping function，a simplified method of mapping function in the process of solving complex stress function is given. The results show that：(1) The proposed mapping function method can solve the difficulty caused by horizon，and the existing research results can be applied to the solution of the function coefficients directly. (2) The tunnel boundary mapping accuracy is very high，the relative error can be controlled within 0.5%，the surface boundary mapping accuracy is slightly worse，but all can be controlled within 5%，and the mapping accuracy increases with the increase of the number of coefficients，which can meet the needs of engineering and research. (3) Taking a shallow buried rectangular project as an example，the proposed solution method and simplification method of mapping function are adopted. The results are in good agreement with the finite element results，with a difference of less than 1%.