This paper investigates the numerical solution of a Cauchy problem for harmonic functions modeled by the Fredholm integral equation, In the context of solving this problem, we explore the formulation of the Fredholm equation and apply a regularized Fourier-based method to approximate the solution. The study first introduces the problem setup and the integral equation formulation, then describes the numerical methods employed to solve it. We analyze the influence of key parameters such as the number of Fourier terms N, the number of quadrature points M, and the regularization parameter \( \alpha \) on the accuracy and convergence of the solution. The findings provide practical insights for solving Fredholm equations using numerical methods and highlight the importance of parameter tuning for achieving accurate and stable solutions.