**MTH ****303: Real Analysis I **** ****(4)**

*Learning Objectives*:

This is an introductory course on analysis for BS-MS mathematics students. The aim of this course is to introduce and develop basic analytic concepts of limit, convergence, integration and differentiation. Students who have taken a first course in calculus are suitable for this course.

*Course Contents*:

Real number system, limit superior, limit inferior, supremum principle, completeness, Cantor set

Sequences and series of functions, uniform convergence and its consequences, space of continuous functions on a closed interval, equicontinuous families, Stone-Weierstrass theorem, Arzela-Ascoli theorem

Taylorâ€™s theorem, power series, radius of convergence, exponential, trigonometric and logarithmic functions

Monotonic functions, functions of bounded variation, rectifiable curves

Riemann-Stieltjes integral, properties of Riemann-Stieltjes integral, differentiation of the integral, fundamental theorem of calculus, integration by parts, Gamma function

*Suggested Books*:

- T. M. Apostol,
*Calculus, Volumes 1 and 2*(2nd edition), Wiley Eastern, 1980 - W. Rudin,
*Principles of Mathematical Analysis (3rd Edn.),*McGraw Hill, 1953 - T. M. Apostol,
*Mathematical Analysis (2nd Edn.),*Narosa Publishing, 1985 - R. R. Goldberg,
*Methods of Real Analysis* - H. L. Royden,
*Real Analysis (3rd Edn.),*Prentice Hall, 2008 - Terrance Tao,
*Analysis I & II,*TRIM Series, Hindustan Book Agency

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