M2: Analysis III - Integration - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Ben Green
Course Term: 
Trinity
Course Lecture Information: 

8 lectures

Course Overview: 

In these lectures we define a simple integral and study its properties; prove the Mean Value Theorem for Integrals and the Fundamental Theorem of Calculus. This gives us the tools to justify term-by-term differentiation of power series and deduce the elementary properties of the trigonometric functions.

Learning Outcomes: 

At the end of the course students will be familiar with the construction of an integral from fundamental principles, including important theorems. They will know when it is possible to integrate or differentiate term-by-term and be able to apply this to, for example, trigonometric series.

Course Syllabus: 
Course Synopsis: 

Step functions, their integral, basic properties. Minorants and majorants of bounded functions on bounded intervals. Definition of Riemann integral.

The application of uniform continuity to show that continuous functions are Riemann integrable on closed bounded intervals; bounded continuous functions are Riemann integrable on bounded intervals.

Elementary properties of Riemann integrals: positivity, linearity, subdivision of the interval. The Mean Value Theorem for Integrals. The Fundamental theorem of Calculus; integration by parts and by substitution.

The interchange of integral and limit for a uniform limit of continuous functions on a bounded interval. Term-by-term integration and differentiation of a (real) power series (interchanging limit and derivative for a series of functions where the derivatives converge uniformly).

Reading List: 

Lecture notes will be provided

Further Reading: 

1) W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, Third Edition, 1976).

This is a more advanced text containing more material than is in the course, including the Stieltjes integral.