Introduction to Modern Analysis: Math 523H, Spring 2016

Class Meeting : TuTh 1:00--2:15, LGRT 141

Instructor : Luc Rey-Bellet

Office :  1423 J LGRT
Phone :  545-6020
E-Mail :   luc <at> math.umass.edu
Homepage:   http://www.math.umass.edu/~lr7q
Office Hours :   M 1:30-3:00   Tu 2:30--3:45,   or by appointment.

Syllabus: This course is the first part of an introduction to Analysis (the sequence Math 523 and Math 524). At its core analysis deals with mathematical objects such as functions, limits, sequences, and series and integrals. The ideas and objects in analysis play an important role in differential equations, probability, numerical analysis, geometry, and in most areas of applied mathematics. On the other hand in doing analysis will learn that mathematics is much more than just methods that work. Students will be asked to construct proofs, sometimes long one, and this will open the door to a much deeper understanding of what the language of mathematics really is. In Math 523 we deal mostly with "classical analysis" which is a set of ideas and objects dating back to the 19th century (and earlier of course). Modern analysis (roughly speaking 20th century) objects will appear in Math 524. Topics covered in this class are

• Sequences, limits, accumulations points, series . This is a central object in analysis and in studying it we will revisit the concept of real number. One the highlight is the concept of completeness. We also study series (i.e. infinite sums) or real numbers.
  
  
• Continuity. We introduce the concepts of a continuous function and of uniform continuity and then combine this with the concept of limits to study limits of function.
  
  
• Sequences and series of functions. We combine both previous topics to study power series and introduce the concept of uniform convergence.
  
  
• Differentiation. We study differentiation, prove the mean value theorem, L'Hospital theorem and Taylor expansion of a function.
  
  
• Integration. We study (Rieman) integrals and the fundamental theorem of calculus.
  
  

Textbooks and references: We will follow the textbook only loosely. There are many textbooks to chose from on the topic. The official textbook is suitable for "beginners". It is highly recommended you borrow another text and spend some time reading through it.

Official Textbook:

• Elementary Analysis: The Theory of Calculus by Kenneth Ross. 2nd edition Springer-Verlag, 2013. ISBN-13: 000-1461462703 ISBN-10: 1461462703
  
  

Other recommended textbooks:

• Fundamental Ideas of Analysis by Michael Reed, Wiley, 1998. ISBN-13: 978-0471159964 ISBN-10: 0471159964. An excellent, but pricey book.
  
  
• Introduction to Real Analysis by William Trench. This book, which can be freely and legally downloaded HERE
  
  
• Analysis by its History by by Ernest Hairer and Gerhard Wanner. Springer, 1996. ISBN-13: 978-0387770314 ISBN-10: 0387770313. A wonderful book with a wider scope and a wealth of historical information. The material we covered is essentially Chapter 3 of the book. Highly recommended.
  
  

Grading and assignments: Your grade will be based on homework's (%30), a mid-term exam (%30), a final exam (%30) and class attendance (%10).

• Midterm Exam : Wednesday 3/2, 6:00--8:00 pm, Room TBA
  
  
• Final exam : Friday 4/29, 10:30--12:30 LGRT 141

Homework: Homework will be assigned regularly (essentially every week) and graded.