Specific Objectives of 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.

Course Outline:


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.