Summation of power series by continued exponentials

Abstract

It is proposed that a power series may be summed (analytically continued outside its radius of convergence) by converting it to a continued exponential, which is a structure of the form a0 exp(a1z exp(a2zexp (a3z exp(a4z...)))). The continued-exponential coefficients {ai} for a given function f(z) are determined by equating the Taylor coefficients of the continued exponential with those of f(z). (The coefficients {ai} have a combinatoric interpretation; the nth Taylor coefficient enumerates all n+1-vertex tree graphs whose vertex amplitudes are {ai}.) Continued exponentials have remarkable convergence properties. When a power series has a nonzero radius of convergence, the corresponding continued exponential often converges in a heart-shaped region Ω, whose cusp is determined by the nearest zero or singularity of the function being approximated. The convergence region Ω contains and is much larger than the circle of convergence of the power series. Outside Ω, the complex plane is divided up into an elaborate patchwork of regions in which the continued exponential may either diverge or else approach an N-cycle, N=2,3,4,... .