A common method for obtaining the analytical solution of the unsaturated infiltration Richards equation under the first type of boundary condition is the Laplace transform. By linearizing the differential equation through the Laplace transform, solving it, and then applying the inverse transform, the result can be obtained. In this process, the traditional calculation scheme assumes that the original function converges, which is correct in most engineering cases, but theoretically, counterexamples may exist. This article presents a classic counterexample: when the unsaturated infiltration curve has a power-law form, the original integral diverges, resulting in the non-existence of the Laplace transform and the solution of the equation cannot represent the real infiltration situation. Based on this, this article suggests discussing the convergence of the original function before using the Laplace transform to solve the unsaturated infiltration equation to ensure that the results have mathematical significance. The content discussed in this article can be regarded as an improvement and supplement to the traditional unsaturated infiltration calculation scheme.