CO1: Some of the families and properties of Riemann integrable functions, and the applications of the fundamental theorems of integration.
CO2: Apply Beta and Gamma functions and their properties in finding improper integrals, area under a curve and surface of revolution.
CO3: The valid situations for the inter-changeability of differentiability and integrability with infinite sum, and approximation of transcendental functions in terms of power series.
📝Module:Unit 1 — The Riemann Integral | 14
Definition and existence of Riemann Integral of bounded functions, Darboux theorem, necessary and sufficient condition for R-Integrability, Riemann integrability of continuous functions, monotonic function and function having finite number of discontinuities, Riemann integral as the limit of a sum, fundamental theorem of integral calculus, Mean value theorems.
📝Module:Unit 2 — Improper Integrals and Special Functions | 12
Improper integrals of Type-I, Type-II and mixed type, test for convergence of improper integral such as comparison test and $\mu$-test, Convergence of Beta and Gamma functions and their properties.
📝Module:Unit 3 — Sequences of Functions | 8
Pointwise and uniform convergence of sequence of functions, Cauchy criterion for uniform convergence, theorems on boundedness, continuity, derivability and integrability of the limit function of a sequence of functions with uniform convergence.
📝Module:Unit 4 — Series of Functions | 8
Series of functions, Theorems on the continuity, integrability and derivability of the sum function of a uniformly convergence series of functions, Cauchy criterion for uniform convergence and Weierstrass M-Test.
📝Module:Unit 5 — Power Series and Approximation | 8
Power series, radius of convergence, Cauchy Hadamard Theorem, Differentiation and integration of power series, Abel's Theorem, Weierstrass Approximation Theorem.
📝Textbooks
Bartle, Robert G., & Sherbert, Donald R. (2015). Introduction to Real Analysis (4th ed.). Wiley India Edition. Delhi.
Ghorpade, Sudhir R. & Limaye, B. V. (2006). A Course in Calculus and Real Analysis. Undergraduate Texts in Mathematics, Springer (SIE). First Indian reprint.
Ross, Kenneth A. (2013). Elementary Analysis: The Theory of Calculus (2nd ed.). Undergraduate Texts in Mathematics, Springer.
Shanti Narayan (2019). Elements of Real Analysis. S. Chand Publication. New Delhi.
Ponnusamy, S. (2012). Foundations of Mathematical Analysis. Birkhäuser / Springer.
Jha, K. K. Advanced Real Analysis. Nav Bharat Prakashan.
Mapa, S. K. (2014). Introduction to Real Analysis. Sarat Book Distributor, Kolkata.
02The Riemann IntegralCH
03Improper Integrals and Special FunctionsCH
04Sequences of Functions and Uniform ConvergenceCH