MA 403 Real Analysis (Autumn 2012)

Lectures: Mon 11.35 am -12.30 pm, Tue 8.30 -9.25 am, Thu 9.30 -10.25 am; Tutorial: Thu 3:30 -4.25 pm in Room ICT 01

Informations about the course:

Content:

Review of basic concepts of real numbers: Archimedean property, Completeness.

Metric spaces, compactness, connectedness, (with emphasis on Rn). Continuity and uniform continuity. Derivatives of functions

and Taylor's theorem. Monotonic functions, Functions of bounded variation, Absolutely continuous functions.

Riemann integral and its properties, characterization of Riemann integrable functions. Improper integrals, Gamma functions.

Sequences and series of functions, uniform convergence and its relation to continuity, differentiation and integration.

Fourier series, pointwise convergence, Fejer's theorem, Weierstrass approximation theorem.

Office Hour:

Tue 10.30-11.30 am

Text/References:

1. T. Apostol, Mathematical Analysis, 2nd ed., Narosa Publishers, 2002.

2. R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 2nd ed. 1994.

3. S. R. Ghorpade and B. V. Limaye, A Course in Calculus and Real Analysis, Springer, 2006.

4. W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1983.

Assignments

2. From Apostol exercises: 3.2 a, b, f, g; 3.3 d, e, f; 3.5; 3.27; 3.29; 3.30; 3.31; 3.32

3. From Apostol exercises: 3.12 a, b; 3.13; 3.15; 3.16; 3.17; 3.20; 3.41; 3.42

5. From Apostol exercises: 4.10; 4.13; 4.18; 4.28 c, d, e, f, g; 4.30, 4.51, 4.53, 4.56, 4.58

6. From Apostol exercises: 8.15 a, c, d, e, f, g, h, k, l; 8.24; 8.25

8. From Apostol exercises: 6.1, 6.2

9. From Apostol exercises: 9.2; 9.5; 9.7; 9.14 c; 9.15; 9.16; 9.19; 9.22; 9.31 b, c