Syllabus
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Course Code: MMATH20-104 Course Name: Real Analysis |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Definition and existence of the Riemann-Stieltjes integral, properties of the integral, integration and differentiation, the fundamental theorem of calculus, integration of vector-valued functions, rectifiable curves. (Scope as in Chapter 6 of ‘Principles of Mathematical Analysis’ by Walter Rudin, Third Edition). | |
| 2 | Sequences and series of functions: Pointwise and uniform convergence of sequences of functions, Cauchy criterion for uniform convergence, Dini’s theorem, uniform convergence and continuity, uniform convergence and Riemann integration, uniform convergence and differentiation. (Scope as in Sections 9.1 to 9.3 of Chapter 9 ‘Methods of Real Analysis’ by R.R. Goldberg). Convergence and uniform convergence of series of functions, Weierstrass M-test, integration and differentiation of series of functions, existence of a continuous nowhere-differentiable function, the Weierstrass approximation theorem (Scope as in Sections 9.4, 9.5, 9.7 of Chapter 9 & Section 10.2 of Chapter 10 of ‘Methods of Real Analysis’ by R.R. Goldberg). |
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| 3 | Functions of several variables: Linear transformations, the space of linear transformations on Rn to Rm as a metric space, open sets, continuity, derivative in an open subset of Rn, chain rule, partial derivatives, continuously differentiable mappings, the contraction principle, the inverse function theorem, the implicit function theorem. (Scope as in relevant portions of Chapter 9 (up to 9.29) of ‘Principles of Mathematical Analysis’ by Walter Rudin, Third Edition) | |
| 4 | Power Series: Uniqueness theorem for power series, Abel’s and Tauber’s theorem, Taylor’s theorem, Exponential & Logarithmic functions, trigonometric functions, Fourier series, Gamma function (Scope as in relevant portions of Chapter 8 of ‘Principles of Mathematical Analysis’ by Walter Rudin, Third Edition). | |
