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With the objective of identifying intrinsic forms of mathematical production in complex analysis (CA), this study presents an analysis of the mathematical activity of five original works that contributed to the development of Cauchy's integral theorem. The analysis of the mathematical activity was carried out through the identification of the types of expressions used and the way they were used by the historical subjects when communicating their results, to subsequently identify transversal elements of knowledge production. The analysis was refined by the notion of confrontation, which depicts the development of mathematical knowledge through the idea of building knowledge against previous knowledge. As a result of the study we established epistemological hypothesis, which are conceived as conjectures that reveal ways in which mathematical knowledge was generated in CA.

This study addresses a question about the similarities and differences between original mathematical works in complex analysis and contemporary textbooks, regarding the use of figures (conceived as two-dimensional drawings) to address concepts in this branch of mathematics. In order to answer this question, we analyzed the four main textbooks that are referenced in the teachers' guides of all Spanish public universities that offer a degree in mathematics. Specifically, we present how these four textbooks structure the concept of complex integral and the proof of Cauchy's integral theorem. To carry out our analysis, we retrieved a reference epistemological model that describes how historical subjects used figures to develop complex analysis from the first quarter of the 19th century to the first half of the 20th century. The study shows that the four textbooks structure these concepts in such a way that they are related to the most contemporary forms in which they were attended in their historical development. Although, we are not against structuring the content of the textbooks in this way, we will argue how the reference epistemological model can serve as an epistemological alternative for the elaboration of didactic material that takes into account a historical development of complex analysis. [ABSTRACT FROM AUTHOR]