This communication deals with the analytical solutions of the Cauchy problem for the Cauchy–Riemann system of equations, which is generally unstable according to Hadamard, but its solution exists if the initial data are analytic. Here, we use the Vectorial Adomian Decomposition (VAD) method, Vectorial Variational Iteration (VVI) method, and Vectorial Modified Picard’s (VMP) method to obtain the convergent series solution. The proposed schemes provide analytical approximations in infinite series form without requiring discretization. These methods are efficient and reliable, as demonstrated through six model problems with various source terms and analytic initial data.