Aspects and images of complex path integrals. An epistemological analysis and a reconstruction of experts' interpretations of integration in complex analysis

Abstract

The first research reports from complex analysis education show that not only novices but also mathematical experts have difficulties in interpreting complex path integrals. Therefore, we deal with two sides of experts' complex analysis discourse in this thesis: On the one hand, we present a comprehensive epistemological analysis of complex path integrals. On the other hand, we reconstruct experts' personal interpretations of these mathematical objects in the form of a multi case study. The thesis has three major contributions, which are grounded theoretically in the commognitive framework and German subject-matter didactics:

We suggest a conceptualisation of discursive mental images as narratives and discursive frames as sets of metarules in intuitive mathematical discourses in order to enrich basic research in mathematics education at university level. It complements acquisitionist perspectives on individuals' mental images of mathematical objects and provides a non-subsumptive and non-prescriptive way to study experts' individual, intuitive interpretations of mathematical objects. A detailed, historically informed epistemological analysis of definitions of complex path integrals, their discursive embedding, and curricular connections to other mathematical discourses enables us to identify four so-called aspects and four partial aspects of complex path integrals. These are typical ways of defining complex path integrals by relating them to different mathematical constraints on the integrands, paths, or domains. We also provide a new axiomatic definition for complex path integrals of holomorphic functions. This conceptualisation from the first part is used for the analysis of experts' intuitive mathematical discourses about complex path integrals. Our study also includes their individual interpretations and substantiations of central integral theorems in complex analysis.

The reconstructed set of discursive images contains an analogy-based saming of complex and real path integrals, the valuation of the complex path integral as a tool, a mean value interpretation, and others. One expert also attempted to transfer area interpretations for real integrals to complex path integrals. In particular, experts' intuitive interpretations of complex path integrals are primarily narrative rather than figurative. The theoretical construct of discursive frame turns out to be especially helpful as it enables us to highlight commonalities and differences between experts' intuitive mathematical discourses about complex path integrals. Consistent with previous literature, this study confirms that experts enrich their intuitive mathematical discourses with connections to other mathematical discourses such as real or vector analysis. We conclude with perspectives for future research on the teaching and learning of complex path integrals.