Euler's Series for Sine and Cosine. An Interpretation in Nonstandard Analysis

2022/09/08 by Piotr Błaszczyk, Błaszczyk, Piotr, Anna Petiurenko +1
Arts and Humanities · Mathematics · #FOS: Mathematics #Historical Philosophy and Science #History and Overview (math.HO) #History and Theory of Mathematics #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2304.01353

openalex publication_date 2022/09/08 · openalex created_date 2023/04/07 · openalex updated_date 2026/07/28

Abstract

In chapter VIII of Introductio in analysin infinitorum, Euler derives a series for sine, cosine, and the formula e^iv=cos v+isin v His arguments employ infinitesimal and infinitely large numbers and some strange equalities. We interpret these seemingly inconsistent objects within the field of hyperreal numbers. We show that any non-Archimedean field provides a framework for such an interpretation. Yet, there is one implicit lemma underlying Euler's proof, which requires specific techniques of non-standard analysis. Analyzing chapter III of Institutiones calculi differentialis reveals Euler's appeal to the rules of an ordered field which includes infinitesimals -- the same ones he applies deriving series for sin v, cos v, and e^v.

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