On the Functional Partial Inversion: Theory and Potential Applications

This paper introduces the theory of Functional Partial Inversion (FPI), a novel framework that constructs a continuous spectrum of operators bridging the identity map and a function’s classical inverse. We define a one-parameter family f [α] for a bijective function f , where the degree of inversion α ∈ [0, 1] governs the interpolation: α = 0 yields the identity operator, and α = 1 yields the standard inverse f −1. We present three constructive schemes (additive, multiplicative, and resolvent) that satisfy a consistent set of axioms. The paper establishes key theorems on existence, uniqueness, and flow dynamics for each scheme, proving that FPI families are well-defined for broad classes of functions. The primary utility of FPI is demonstrated in its applications, most notably as a homotopy-based stabilization tool for solving transcendental equations, where α evolves from 0 to 1 to ensure robust global convergence. Additional applications in cryptography, data privacy, regularized signal deconvolution, and dynamical systems are outlined. Our aim is to establish FPI as a versatile construct with significant interdisciplinary potential.