On the analytic inversion of functions, solution of transcendental equations and infinite self-mappings

The solution of seemingly simple transcendental equations is in effect equivalent to the general problem of analytical inversion of functions. Within a powerful and systematic method, based on the solution of an associated Riemann–Hilbert boundary value problem, beautiful explicit results for various inverse functions of physical importance have been found which inevitably take on the guise of integral representations of these functions. In an attempt to reduce one such solution to a standard-function expression which would then be easy to evaluate, we recognize an infinite ladder self-mapping solution. This new perspective, born out of complicated complex analysis, is straightforwardly and uniquely related to the systematic generation of fast converging expansions within the corresponding regions of single-valuedness of the inverse function.