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Communications of the ACM

Communications of the ACM

On a class of iteration formulas and some historical notes


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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.

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