A Modern Introduction to Mathematical Analysis

This textbook presents all the basics for the first two years of a course in mathematical analysis, from the natural numbers to Stokes-Cartan Theorem.
The main novelty which distinguishes this book is the choice of introducing the Kurzweil-Henstock integral from the very beginning. Although this approach requires a small additional effort by the student, it will be compensated by a substantial advantage in the development of the theory, and later on when learning about more advanced topics.
The text guides the reader with clarity in the discovery of the many different subjects, providing all necessary tools no preliminaries are needed. Both students and their instructors will benefit from this book and its novel approach, turning their course in mathematical analysis into a gratifying and successful experience.

Features an original approach to the exponential and circular functions Explains all the main analysis theory in only 400 pages Presents the Kurzweil-Henstock integral

Auteur

Alessandro Fonda obtained his PhD at the International School for Advanced Studies (SISSA-Trieste) in 1988, under the supervision of Jean Mawhin. Then he first got a research position in Belgium, before going back to Italy, at the University of Trieste, becoming full professor in 2002.He obtained two prizes from the Académie Royale de Belgique for his monographs "Periodic solutions of scalar second order differential equations with a singularity" in 1993 and "Playing around resonance. An invitation to the search of periodic solutions for second order ordinary differential equations" in 2017.

He has been invited to give plenary lectures at several conferences in Europe and in the USA.

His research interests are mainly concentrated on existence and multiplicity problems for boundary value problems. He recently gave an important contribution to the study of the periodic problem for Hamiltonian systems by providing some generalizations of the Poincaré-Birkhoff Theorem in higher dimensions. He also studied problems related to differential inclusions, persistence in dynamical systems, and global behaviour of the solutions to planar systems.

Résumé

"A Modern Introduction to Mathematical Analysis, by Alessandro Fonda, is a well-written text that provides an enjoyable narrative of the major results of (advanced undergraduate or elementary graduate) analysis. ... The text is self-contained ... . I would recommend it to anyone needing a refresher on the fundamental results and foundations in Analysis, or for those looking for a different perspective on the material." (John Ross, MAA Reviews, January 28, 2024)

Contenu

• Part I The Basics of Mathematical Analysis. -
  1. Sets of Numbers and Metric Spaces. - 2. Continuity. - 3. Limits. - 4. Compactness and Completeness. - 5. Exponential and Circular Functions. - Part II Differential and Integral Calculus in R. -
     6. The Derivative. - 7. The Integral. - Part III Further Developments. -
        8. Numerical Series and Series of Functions. - 9. More on the Integral. - Part IV Differential and Integral Calculus in RN. - 10.The Differential. - 11. The Integral. - 12. Differential Forms.