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In this study, the impact of principal stress states on the stress characteristics and initial failure of the rock mass surrounding a three-center arch opening was investigated using complex variable function methods and Discrete Element Method (DEM) numerical modeling. First, the mapping function of the opening was determined using the trigonometric interpolation method, and the influence of the number of terms in the mapping function on its accuracy was revealed. Based on this, the far-field stress state of the underground rock mass was characterized by the ratio of the minimum to maximum principal stress (λ) and the angle (β) between the principal stress and the vertical direction. This stress state was then converted into normal and shear stresses. Using complex variable function theory, the stress characteristics at the boundary of the opening under different stress states were analyzed. Finally, DEM numerical modeling was employed to study the initial failure characteristics at the boundary of the opening and its relationship with the stress distribution. The results indicate that the lateral pressure coefficient significantly affects the stability of the opening by influencing stress concentration around the surrounding rock. Low lateral pressure coefficients lead to tensile stress concentration at the boundary perpendicular to the maximum principal stress. As the coefficient increases, tensile stress decreases, and compressive stress areas expand. While the principal stress direction has a minor effect on stress concentration, it notably impacts stress distribution at the boundary. When λ < 1.0 and β = 45°, stress distribution asymmetry is most pronounced, with the highest compressive stress. The early failure distribution aligns with stress concentration areas, validating the use of stress analysis in predicting opening stability and failure characteristics.