B.Sc. Mathematics
3rd Semester Syllabus
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MJC-03 : Real Analysis
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Unit 1: Real Number System
◦Dedekind theory of real numbers
◦Algebraic and order properties of R
◦Archimedean Property
◦Density Theorem
◦Completeness property of R
◦Bounded sets
◦Theorems on Suprema and Infima
Unit 2: Topology of Real Numbers
◦Neighbourhood of a point in R
◦Open and closed sets
◦Limit points and isolated points of a set
◦Bolzano-Weierstrass theorem for a set
◦Derived set
◦Clouser and Interior of a set
Unit 3: Sequences
◦Sequence and its convergence
◦Bounded sequence
◦Monotone sequences
◦Subsequences
◦Limit of a sequence
◦Limit Theorem
◦Bolzano-Weierstrass theorem for sequences
◦Limit superior and limit inferior for bounded sequence
◦Cauchy sequence
◦Cauchy's general principle of convergence
Unit 4: Infinite Series
◦Infinite series and their convergence
◦Cauchy Criterion
◦Tests for convergence: Comparison test
◦D'Alembert Ratio Test
◦Cauchy's root test
◦Rabbe's test
◦Logarithmic test
◦D'Morgan and Bertrand's test
◦Cauchy integral test
◦Cauchy condensation test
◦Gauss's test
◦Alternating series
◦Leibnitz test
◦Absolute and Conditional convergence
Reference Books:
- Bartle, Robert G., & Sherbert, Donald R. (2015). Introduction to Real Analysis (4th ed.). Wiley India Edition. New Delhi
- Ross, Kenneth A. (2013). Elementary Analysis: The theory of calculus (2nd ed.). Undergraduate Texts in Mathematics, Springer. Indian Reprint
- Malik, S. C. & Arora, Savita. (2021). Mathematical Analysis (6th ed.). New Age International Publishers, New Delhi
- Jha,K.K. Advanced Course in Real Analysis and Higher Analysis. New Bharat Publishing House
- Simmons, George F. (2016). Differential Equations with Applications and Historical Notes. Tata- McGraw Hill Publishing Company Limited, New Delhi
- Raisinghania, M.D. (2020). Ordinary and Partial Differential Equations (20th ed.). S. Chand Publication
- Bronson, R. &Coasta, Gabriel B. (2021). Schaum's Outline of Differential Equations (5th ed.). McGraw Hill
- Prasad, Lalji. (2019). Differential Equations. Paramount Publication