Lec-01 Riemann Integral. Upper and Lower Riemann Sums. Real II. Rudin (Urdu/Hindi)

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In this part of lecture series course "Real Analysis II" Course of BS mathematics 5th Semester, we shall cover the following topics.

The Riemann-Stieltjes Integrals
x Definition and existence of integrals
x Properties of integrals
x Fundamental theorem of calculus and its applications
x Change of variable theorem
x Integration by parts
Functions of Bounded Variation
x Definition and examples
x Properties of functions of bounded variation
Improper Integrals
x Types of improper integrals
x Tests for convergence of improper integrals
x Beta and gamma functions
x Absolute and conditional convergence of improper integrals
Sequences and Series of Functions
x Power series
x Definition of point-wise and uniform convergence
x Uniform convergence and continuity
x Uniform convergence and integration
x Uniform convergence and differentiation
x Examples of uniform convergence


Recommended Books
1. W. Rudin, Principles of Mathematical Analysis, 3rd edition, (McGraw Hill, 1976)
2. R. G. Bartle, Introduction to Real Analysis, 3rd edition, (John Wiley and Sons, 2000)
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