Mathematics : Function representations - 3PUKMAT1

Goal(s)

This module aims to introduce tools and methods related to the scientific approach. It covers models, measurement, representation spaces, and introduces how to refute models (or, by default, refrain from such refutation). Finally, the whole is compiled in the form of a report of its experimental results.

We closely follow what George Box (1976) wrote and what has been very well summarized at https://blogs.sas.com/content/iml/2025/04/02/all-models-are-wrong.html (in English, translation below). The author's analysis is based on “Science and Statistics” (Box, 1976, JASA, Vol. 71, pp. 791-799).

Box discusses the scientific method as an iterative process: practice meets theory and theory meets practice. Scientific advances require the production of parsimonious but effective models that will be compared with data to expose their imperfections.

Since all models are wrong, scientists (engineers) cannot obtain a correct model but must seek an economical (parsimonious) description of the natural phenomenon. Above all, scientists (engineers) should be able to rule out completely inadequate models.

To paraphrase M. Box: "In applying mathematics to physics or statistics, we attempt to make assumptions about the world around us. We know they are false, but it can be argued that certain

Content(s)

Representations of functions

• Orthogonal polynomials: scalar products, solving ordinary linear differential equations
• Fourier series: integration of a complex function over a part of IR, properties, solving ordinary linear differential equations with periodicity conditions
• Fourier transform: integration of a complex function on IR, properties, solving ordinary linear differential equations on IR, introduction to distributions or generalized functions
• Unilateral Laplace transform: integration of a complex function on IR+, properties, solving ordinary linear differential equations with initial conditions
• Inverse Laplace transform (Mellin-Fourier integral): integration of a complex function over a subset of the complex set, complex integration theory, residue theorem

Two-year college-level calculus course (limits and continuity, series, integration, differentiation, limit developments)

Test

Additional Information

Bibliography

• Walter Appel : Mathématiques pour la physique et les physiciens, 5°éd., 608 pages, Éditions H&K, ISBN: 978-2-35141-339-5 (2017) [exemplaires disponibles dans la bibliothèque de Phelma].
• Gilles Pages, Marc Briane : Analyse - Théorie de l'intégration (Convolution, Transformées de Fourier et de Laplace), 8°éd., 432 pages, deBoeck supérieur, ISBN 978-2-8073-5955-0 (2023).
• Mourad Choulli : Analyse complexe, 1°éd, 192 pages, deBoeck supérieur, ISBN 978-2-8073-2749-8 (2020).