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Default Kashmir University M.Sc. Methematics 1st Sem MMCP102 REAL ANALYSIS Syllabus

Kashmir University M.Sc. Methematics 1st Sem MMCP102 REAL ANALYSIS Syllabus

Semester – I REAL ANALYSIS Course No. MM-CP-102 Maximum Marks: 100 Duration of Examination: 3 hrs (a) External Exam: 80 (b) Internal Exam: 20

Unit I Integration : Definition and existence of Riemann – Stieltjes integral , behavior of upper and lower sums under refinement , Necessary and sufficient conditions for RS-integrability of continuous and monotonic functions , Reduction of an RS-integral to a Riemann integral , Basic properties of RS-integrals , Differentiability of an indefinite integral of a continuous functions , The fundamental theorem of calculus for Riemann integrals .

Unit II

Improper Integrals: Integration of unbounded functions with finite limit of integration. Comparison tests for convergence, Cauchy’s test. Infinite range of integration. Absolute convergence. Integrand as a product of functions. Abel’s and Dirichlet’s test. Inequalities: Arithmetic-geometric means equality, Inequalities of Cauchy Schwartz, Jensen, Holder& Minkowski. Inequality on the product of arithmetic means of two sets of positive numbers.

Unit III Infinite series: Carleman’s theorem. Conditional and absolute convergence, multiplication of series, Merten’s theorem, Riemann’s rearrangement theorem. Uniform continuity, Heine’s theorem on uniform continuity, Young’s form of Taylor’s theorem, generalized second derivative. Darboux theorem on the derivative. Bernstein’s theorem and Abel’s limit theorem.

Unit IV Sequence and series of functions: Point wise and uniform convergence, Cauchy criterion for uniform convergence, Mn - test , Weiestrass M-test , Abel’s and Dirichlet’s test for uniform convergence , uniform convergences and continuity ,R- integration and differentiation, Weirstrass Approximation theorem. Example of continuous nowhere differentiable function.

Reference 1. R. Goldberg : Methods of Real Analysis 2. W.Rudin : Principles of Mathematical Analysis 3. J.M.Apostol : Mathematical Analysis 4. S.M.Shah and Saxena: Real Analysis 5. A.J.White :Real Analysis , An Introduction 6. L.Royden :Real Analysis
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