# MTH322: Real Analysis II (Spring 2016)

This course was teach to MSc III and IV.

## Course Contents:

**Sequences of functions:** convergence, uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation, the exponential and logarithmic function, the trigonometric functions.

**Series of functions:** Absolute convergence, uniform convergence, Cauchy criterion, Weiestrass M-test, power series of functions, radius of convergence, Cauchy-Hadamard theorem, differentiation theorem, uniqueness theorem.

**Improper integrals:** Improper integral of first and second kind, comparison tests, Cauchy condition for infinite integrals, absolute convergence, absolute convergence of improper integral, uniform convergence of improper integrals, Cauchy condition for uniform convergence, Weiestrass M-test for uniform convergence.

## Notes & handout

- Summary: Riemann-Stieltjes Integral (485.16 KiB, 584 downloads) | View Online
- Chapter 01: Improper Integral of 1st and 2nd Kinds (738.69 KiB, 656 downloads) | View Online
- Chapter 02: Functions Defined by Improper Integrals (528.3 KiB, 459 downloads) | View Online
- Chapter 03: Sequences and Series of Functions (143.59 KiB, 485 downloads) | View Online
- Questions from last chapter (77.64 KiB, 227 downloads) | View Online

## Recommended Books

- Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner, Elementary Real Analysis:Second Edition (2008) URL: http://classicalrealanalysis.info/Elementary-Real-Analysis.php
- Rudin, W. (1976). Principle of Mathematical Analysis, McGraw Hills Inc.
- Bartle, R.G., and D.R. Sherbert, (2011): Introduction to Real Analysis, 4th Edition, John Wiley & Sons, Inc.
- Apostol, Tom M. (1974), Mathematical Analysis, Pearson; 2nd edition.
- Somasundaram, D., and B. Choudhary, (2005) A First Course in Mathematical Analysis, Narosa Publishing House.