MTH322: Real Analysis II (Spring 2022)

This course is offered to BS, Semester VI at Department of Mathematics, COMSATS University Islamabad, Attock campus. This course need rigorous knowledge of continuity, differentiation, integration, sequences and series of numbers, that is many notion included in Real Analysis I.

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.

• Riemann Integrals: Summary | Download PDF

• Ch 01: Improper Integrals of 1st and 2nd Kinds | Download PDF

• Review: Sequences and Series | Download PDF

• Ch 02: Sequences and Series of Functions | Download PDF

• Assignment 01 | Downlead PDF | View Online

• Assignment 02 | Downlead PDF | View Online

Please click on View Online to see inside the PDF.

• https://www.mathsisfun.com/numbers/infinity.html
• http://www.sosmath.com/calculus/improper/absconv/absconv.html
• http://world.mathigon.org/Sequences
• http://www.math.wisc.edu/~park/Fall2011/integration/Improper%20Integral.pdf
• https://www.quora.com/What-is-the-difference-between-continuous-and-uniformly-continuous-for-a-function
• http://www.personal.psu.edu/auw4/M401-lecture-notes.pdf

1. Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner, Elementary Real Analysis:Second Edition (2008) URL: http://classicalrealanalysis.info/Elementary-Real-Analysis.php
2. Rudin, W. (1976). Principle of Mathematical Analysis, McGraw Hills Inc.
3. Bartle, R.G., and D.R. Sherbert, (2011): Introduction to Real Analysis, 4th Edition, John Wiley & Sons, Inc.
4. Apostol, Tom M. (1974), Mathematical Analysis, Pearson; 2nd edition.
5. Somasundaram, D., and B. Choudhary, (2005) A First Course in Mathematical Analysis, Narosa Publishing House.
6. http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
7. S.C. Malik and S. Arora, Mathematical analysis, New Age International, 1992. (Online google preview)