Asymptotic and essentially singular solutions of the Feigenbaum equation

For suitably defined large N, we express Feigenbaum's equation as a singular Schroder functional equation whose solution is obtained using a scaling ansatz. In the limit of infinite N certain self-consistency conditions on the scaled Schroder solution lead to an essentially singular solution of Feigenbaum's equation with a length scale factor of ..cap alpha.. approx. = 0.0333 and a limiting feigenvalue of delta/sub infinity/approx. = 30.50, in agreement with Eckmann and Wittwer's values of ..cap alpha.. = 0.0333831... and their conjectured estimate of delta/sub infinity/ less than or equal to 30.