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Indian Institute of Science Education and Research Bhopal

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

Taylor’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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