====== MTH322: Real Analysis II (Fall 2020) ====== This course is offered to MSc, Semester II 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 [[atiq:sp20-mth321|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 Integrals | {{ :atiq:fa20-mth322-review-riemann-sum.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=fa20-mth322-review-riemann-sum#online_view|View Online]] * Ch01: Improper Integrals of 1st and 2nd Kinds | {{ :atiq:fa20-mth322-ch01.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=fa20-mth322-ch01#online_view|View Online]] * Review: Sequence & Series | {{ :atiq:fa20-mth322-pre-ch02.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=fa20-mth322-pre-ch02#online_view|View Online]] NEW * Ch02: Sequences and Series of Functions | {{ :atiq:fa20-mth322-ch02.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=fa20-mth322-ch02#online_view|View Online]] NEW ===Assignments and Quizzes=== * Assignment 01 | {{ :atiq:fa20-mth322-assignment-01.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=fa20-mth322-assignment-01#online_view|View Online]] * Assignment 02 | {{ :atiq:a2-mth322.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=a2-mth322#online_view|View Online]] NEW * Quiz 01 | {{ :atiq:fa20-mth322-quiz-01.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=fa20-mth322-quiz-01#online_view|View Online]] * Quiz 02 | {{ :atiq:q2-mth322.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=q2-mth322#online_view|View Online]] * Quiz/Assignment 04 | {{ :atiq:qa4-mth322.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=qa4-mth322#online_view|View Online]] * Quiz/Assignment 04 (Revised) | {{ :atiq:qa4-revised-mth322.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=qa4-revised-mth322#online_view|View Online]] * Quiz/Assignment 04 (Revised2) | {{ :atiq:qa4-rev2-mth322.pdf |Download PDF}} | VIEW [[:atiq:fa20-mth322?f=qa4-rev2-mth322#online_view|View Online]] NEW {{include>atiq-notes-viewer.php}} ===Videos===
===Online resources=== * 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 =====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. - http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF - S.C. Malik and S. Arora, Mathematical analysis, New Age International, 1992. ([[https://books.google.com.pk/books?id=L25yJhCll78C&lpg=PR1&ots=ZWf6qZ1C7y&dq=S.C.%20Malik%20and%20S.%20Arora%2C%20Mathematical%20analysis%2C%20New%20Age%20International%2C%201992.&pg=PP1#v=onepage&q=S.C.%20Malik%20and%20S.%20Arora,%20Mathematical%20analysis,%20New%20Age%20International,%201992.&f=false|Online google preview]])