Limits of nonlinear weak asymptotic methods

We propose a mathematical limit of $L^1$-stable weak asymptotic methods. A family of $L^1$-stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex domain having the real space on its boundary. This provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations. The weak limit of the approximate solutions from weak asymptotic methods in the space of bounded Radon measures is recovered as a boundary value of this holomorphic function.