Experts’ intuitive mathematical discourses about integration in complex analysis

Abstract

Although complex analysis is part of the study programs of many mathematics undergraduates, little research has been done on how individuals interpret basic concepts from complex analysis. To address this gap, this paper investigates how experts individually think about complex path integrals. For this purpose, the commognitive framework is used to conceptualize experts’ interpretations of mathematical concepts discursively, namely in terms of so-called intuitive mathematical discourses. A total of nine interpretations of complex path integrals, so-called discursive images, as well as eight sets of rules governing their construction, so-called discursive frames, are derived from expert interviews. These interpretations range from a rejection of intrinsic meaning to connections with real and vector analysis, mean values, and individual formulations of theorems. The paper also raises questions for the inclusion of the results into teaching and addresses further research.