DESCRIPTION AND OBJECTIVES

A continuation of Real Analysis I, this course will continue to cover the fundamentals of real analysis, concentrating on the Riemann-Stieltjes, Riemann integrals, Functions of Bounded Variation, Improper Integrals, and convergence of series. Emphasis would be on proofs of main results.The aim of this course is also to provide an accessible, reasonably paced treatment of the basic concepts and techniques of real analysis for students in these areas.

Pre-Requisite

Real Analysis-I

INTENDED LEARNING OUTCOMES

1. Determine the Riemann inerrability and the Riemann-Stieltjes integrability of a bounded function and prove a selection of theorems concerning
2. integration,
3. Recognize the difference between point wise and uniform convergence of a sequence of functions,
4. Illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability, and integrability, and
5. Illustrate the convergence properties of power series.

COURSE CONTENTS

1. The Riemann-Stieltjes Integrals:
2. Definition and existence of integrals,
3.  properties of integrals,
4. fundamental theorem of calculus and its applications,
5. Change of variable theorem, integration by parts.
6. Functions of Bounded Variation:
7. Definition and examples, properties of functions of bounded variation.
8. Improper Integrals: Types of improper integrals,
9.  tests for convergence of improper integrals,
10. Beta and gamma functions, absolute and conditional convergence of improper integrals.
11. Sequences and Series of Functions:
12. Power series, definition of point-wise and uniform convergence, uniform convergence and continuity,
13.  Uniform convergence and differentiation, examples of uniform convergence.

Recommended Books

1. Walter Rudin, Principles of Mathematical Analysis, (3rd Ed. 1976)
2. T.M. Apostal, Mathematical Analysis,( 2nd Ed. Addison Wesley, 1974)
3. W. Kaplan, Advanced calculus, (5th Ed. Addison Wesley, 2002)
4. R.L. Rabenstein, Elements of Ordinary differential equations, (Academic Press, 1984)
5. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis,( 3rdEd.1999)
6. James Stewart, Calculus, (8th Ed)

ASSESSMENT CRITERIA

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam:   30

Final-Term Exam: 50

Key Dates and Time of Class Meeting

Wednesday to Thursday: 11:00am -12:30 pm (BS-VI Regular)

Wednwsday:                     3:30pm - 5:00 pm    (BS-VI Self Support) 

 Thursday:                         12:30pm - 2:00 pm    (BS-VI Self Support)

Commencement of Classes: January 13, 2020 (Monday)

Mid Term Examination: March 9-13, 2020 (Monday to Friday)

Final Term Examination: Jun 2020

Declaration of Result: July 2020