Analysis III: Integration: 2008-2009
Term | Trinity Term 2009 (8 lectures) |
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.Synopsis
Step functions, their integral, basic properties. The application of uniform continuity to approximate continuous functions above and below by step functions. The integral of a continuous function on a closed bounded interval. Elementary properties of the integral of a continuous function: positivity, linearity, subdivision of the interval.
The Mean Value Theorem for Integrals. The Fundamental Theorem of Calculus; linearity of the integral, integration by parts and 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); examples to include the derivation of the main relationships between exponential, trigonometric functions and hyperbolic functions.
Reading list
- T. Lyons Lecture Notes.
- J. Roe, Integration Mathematical Institute Notes (1994).
- H. A. Priestley, Introduction to Integration (Oxford Science Publications, 1997), Chapters 1–8. [These chapters commence with a useful summary of background `cont and diff' and go on to cover not only the integration but also the material on power series.]
- Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis (Wiley, Third Edition, 2000), Chapter 8.
Taking our courses
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