Newton method; difference equation; series expansion; fixed point; discrete dynamical system; Julia set; Cayley’s problem; recurrence relations; analytic solution
Newton's method for computation of a square root yields a difference equation which can be solved using the hyperbolic cotangent function. For the computation of the third root Newton's sequence presents a harder problem, which already Cayley was trying to solve. In the present paper two mutually inverse functions are defined in order to solve the difference equation, instead of the hyperbolic cotangent and its inverse. Several coefficients in the expansion around the fixed points are obtained, and the expansions are glued together in the region of overlapping.
 A. Cayley: The Newton-Fourier imaginary problem. Amer. J. Math. II, 97 (1879).
 G. Julia: Sur l'iteration des fonctions rationnelles. Journal de Math. Pure et Appl. 8, 47-245 (1918).
 H. O. Peitgen P. H. Richter: The Beauty of Fractals
. Springer-Verlag, Berlin, Heidelberg 1986. MR 0852695