Inversion of Characteristic Functions Without Imaginary Unit

We start from a pure vector update of the concepts of complex numbers and the characteristic function of a probability distribution, free from the imaginary unit, and update some basic properties of the latter. In particular, we derive vector inverse formulas for probability distributions and density functions, provide a new geometric proof of the convergence of suitably centered and normalized vector-powers of characteristic functions, and thus provide an update to the proof of the central limit theorem. Moreover, we demonstrate how to eliminate the imaginary unit from various equations—such as Schrödinger’s equation—by transitioning to vector equations.