Analysis II

Semester and Year Offered: MS 2018

Course Coordinator and Team: Ramneek Khassa and others

Email of course coordinator: ramneek[at]aud[dot]ac[dot]in

Pre-requisites: Mathematics at the XII grade level and 4 credits of MA502 (Analysis I).

Aim: This course has been designed with the aim to adding to the skills of the students in Calculus and to deepen their understanding. The degree of rigour in considering the definitions and concepts increases and develops students thinking and understanding of concepts.

Course Outcomes: After completing this course, students should be able to

1. Describe fundamental properties of the real numbers.
2. Demonstrate an understanding of limits and how they are used in sequences, series, continuity, differentiation.
3. Construct rigorous mathematical proofs of basic results in real analysis.

Brief description of modules/ Main modules:

The Real Numbers

Algebraic and Order Properties of Real Numbers, The Completeness Properties of Real Number and applications of Supremum Property , Intervals.

Sequences and Series

Review of sequences, Monotone Sequence, Monotone Convergence Theorem, Subsequence and the Bolzano-Weierstass Theorem, Properly Divergence Sequences, Introduction to Infinite Series, Absolute Convergence, Test for Absolute Convergence

Limits and Continuous Functions

Limits of Functions, Limit Theorems, Some Extensions of Limit Concepts, Continuous Functions and Combinations of Continuous Functions, Continuous Functions on Intervals, Uniform Continuity, Weierstrass Approximation Theorem and Its Applications (Statement Only), Monotone and Inverse Functions

Differentiation

The Derivative, The Mean Value Theorem, L’Hospital Rule’s, Cauchy’s Mean Value theorem (without Proof), Taylor’s Theorem

Assessment Details with weights:

Reading List:

• Bartle, R. G. and Sherbert, D. R., Introduction to Real Analysis, 3rd Edition, John & Wiley Sons, 2000.
• Ross, K. N., Elementary Analysis: Review of Calculus, Springer, 2007.