The use of repulsive fixed points to analytically continue certain functions (Q810182)

The purpose of this paper is to provide a method for analytic continuation of a function \[ (1)\quad \lambda (\zeta)=\lim_{n\to \infty}F_ n(\zeta,z),\quad \zeta \in \Delta \] defined in the following way: \[ (2)\quad F_ 1(\zeta,z)=f_ 1(\zeta,z)\text{ and } F_ n(\zeta,z)=F_{n-1}(\zeta,f_ n(\zeta,z))\text{ for } n>1, \] where all \(f_ n(\zeta,z)\) are analytic in both variables in some domain \(S\times D\), \(f_ n(S,D)\subset D\), and \(f_ n\to f\) in \(S\times D\). \(\Delta\subset S\) is a compact region for \(\zeta\) where \(F_ n(\zeta,z)\) converges for \(z\in D\). If \(f_ n(\zeta,z)=a_ n(\zeta)/(b_ n(\zeta)+z)\) for all n, then \(F_ n(\zeta,0)\) are the approximants of the continued fraction \(K(a_ n(\zeta)/b_ n(\zeta)).\) As for continued fractions, one has under mild conditions that the limit \(\lambda\) (\(\zeta\)) exists for \(z\in D\) and is independent of z. Further, the sequence \(F_ n(\zeta,\alpha)\) will normally converge faster to \(\lambda\) (\(\zeta\)) than \(F_ n(\zeta,z)\) for other values of z, where \(\alpha\) is a fixed point (attractor) of f in D. The author gives sufficient conditions for convergence of \(F_ n(\zeta,\alpha)\) for \(\zeta\) in a larger domain than \(\Delta\).