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# MTH322: Real Analysis II (Fall 2016)

**Discussion**at the end of this page.

This course is offered to MSc, Semester III at Department of Mathematics, COMSATS Institute of Information Technology, Attock campus. The is course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in Real Analysis I.

## 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, assignment, quizzes & handout

#### Notes:

- Summary: Riemann-Stieltjes Integral (506.95 KiB, 442 downloads)
- Chapter 01: Improper Integrals of 1st and 2nd Kinds (770.97 KiB, 376 downloads)
- Chapter 02: Functions Defined by Improper Integrals (915.5 KiB, 310 downloads) | View Online
- Chapter 03: Sequences and Series of Functions (143.36 KiB, 177 downloads) | View Online
- Exponential, Logarithmic and Trigonometric Functions: Sample Questions (78.1 KiB, 104 downloads) | View Online

#### Assignments:

- Assignment 1 (340.92 KiB, 316 downloads) | View Online
- Assignment 2 (114.69 KiB, 167 downloads) | View Online
- Assignment 3 (182.76 KiB, 90 downloads) | View Online
- Assignment 3 (102.09 KiB, 73 downloads) | View Online

#### Quizzes:

- Quiz 01 with answers (98.84 KiB, 196 downloads) | View Online
- Quiz 02 with answers (83.67 KiB, 109 downloads) | View Online
- Quiz 03 with answers (81.49 KiB, 82 downloads) | View Online
- Quiz 04 with answers (83.79 KiB, 67 downloads) | View Online

#### Online resources

## 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.
- S.C. Malik and S. Arora, Mathematical analysis, New Age International, 1992. (Online google preview)

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## Discussion