Introduction to course:

A continuation of Real Analysis I, this course will continue to cover the fundamentals of real analysis, concentrating on the Riemann-Stieltjes 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.

Course Prerequisite(s): Real Analysis-I

Learning outcomes: 

1) Determine the Riemann inerrability and the Riemann-Stieltjes integrability of a bounded function and prove a selection of theorems concerning integration,

2) Recognize the difference between point wise and uniform convergence of a sequence of functions.

3) Illustrate the effect of uniform convergence on the limit function with respect to continuity, differentiability, and integrability, and

4)Illustrate the convergence properties of power series.

Contents:

This course covers the following;      

The Riemann-Stieltjes Integrals: 
Definition and existence of integrals,
 properties of integrals, 
fundamental theorem of calculus and its applications, 
change of variable theorem, integration by parts.
Functions of Bounded Variation: 
Definition and examples, properties of functions of bounded variation. 
Improper Integrals: Types of improper integrals,
 tests for convergence of improper integrals, 
beta and gamma functions, absolute and conditional convergence of improper integrals. 
Sequences and Series of Functions: 
Power series, definition of point-wise and uniform convergence, uniform convergence and continuity,
uniform convergence and differentiation, examples of uniform convergence.

Recommended Books 

1)Bartle R. G.  and Sherbert D. R.Introduction to Real Analysis, 4th Ed. John Wiley & Sons, Inc 2011

2)Rudin W., Principles of Mathematical Analysis, (3rd Ed., McGraw-Hill, 1976).

Suggested Books;

Folland G. B. Real Analysis, 2nd Edition, John Wiley and Sons,( New York, 1999).
Hewitt E and Stromberg K. Real and Abstract Analysis, (Springer-Verlag, Berlin Heidelberg New York, 1965).
Lang S., Analysis I, (Addison-Wesley Publ. Co., Reading, Massachusetts, 1968).

Description of system of Evaluation :

Mid Term Exam  30%

Final Term Exam  50%

Sessional 20% ( Attendance 5%, Assignments 5%, Presentation/Quiz 10%)

Key Dates: 

Commencement of Classes: January 13,2020

Mid Term Examination : March 09-13,2020

Final Term Examination: May 04-8,2020

Declaration of Result : May 19,2020

Time Table : BS VI Ex-PPP old 

Monday 3.30- 5 PM     

Friday  11 AM-12.30 PM.

BS VI Ex - PPP new 

Tuesday 8-9.30 AM , 

Wednesday 9.30 - 11 AM.