Abstract / Synopsis
In his article “Mémoire sur les Intégrales Définies, prises entre des limites imaginaires” of 1825, Cauchy proves that the integral of a complex function f defined on complex numbers is independent of the path of integration if the function f remains finite and continuous along the chosen path. In contemporary literature that analyzes the mathematics of Cauchy, the specific mathematical process that allowed Cauchy to prove this result is not explicitly presented. The purpose of this manuscript is to communicate a symbolic-algebraic derivation that justifies the narrative argument used by Cauchy to support his proof, thus hopefully filling this gap. We do this via a careful reading of some of the other original works of Cauchy. Our reconstruction of Cauchy’s mathematical process is framed within a research project in mathematics education that seeks to recover how historical subjects did mathematics in what we now call complex analysis. We hope that our work can serve as an example of how research in mathematics education can recover different ways in which historical subjects did mathematics, ways that are not being explicitly addressed in contemporary literature, but that can serve as a starting point for didactic innovations in complex analysis.
Recommended Citation
José Gerardo Piña-Aguirre & Rosa María Farfán Márquez, "How did Cauchy prove his integral theorem? A Historical Reconstruction for Education Research," Journal of Humanistic Mathematics, Volume 16 Issue 1 (January 2026), pages 188-202. . Available at: https://scholarship.claremont.edu/jhm/vol16/iss1/11
