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Through mathematical examples, we explore the transition from an area where the function is well understood theoretically using perturbation theory (high energies) to a non-perturbative area (low energies), where expansions based on perturbation theory are no longer valid. To construct a description of the function in the nonperturbative region, we use the analytical matching method and the value of the function at zero as a boundary condition. As different functions may produce the same behavior in the perturbative region, we are looking for an answer to the question: how different will the behavior of these functions be in the non-perturbative region.