On a class of iteration formulas and some historical notes

Abstract

The class of iteration formulas obtainable by rational approximations of “Euler's formula” is derived with the corresponding error estimates. Some historical notes on iterative procedures are followed by a derivation of Euler's formula with the associated error estimate in a new notation which simplifies the error estimate and suggests generalizations. The final section considers the Padé approximants to the “Euler polynomial” and shows how a number of known formulas may be derived from this unified approach. There is a short discussion of the “best” formula.