REAL ANALYSIS
Paper Code:
MAT301
Credits:
3
Contact Hours:
45.00
Max. Marks:
100.00
Unit I:
I
9.00
Order completeness of Real numbers, open and closed sets, limit point of sets, Bolzano Weirstrass theorem ,concept of compactness, Heine Borel theorem.
Unit II:
II
9.00
Real Sequences , Limit and convergence of a sequence, Monotonic sequences, Cauchy’s sequences, Sub sequences and Cauchy’s General principle of convergence,
Infinite series and their convergences – Comparison test, Cauchy’s nth root test, D’Alembert, Raabe’s ,Cauchy’s Test, Logarithmic test.
Unit III:
III
9.00
Alternating Series – Leibnitz Test, Absolute and conditional convergence, Properties of continuous function on closed interval, derivable functions:-Derivative of composite function, The inverse function theorem and darboux theorem.
Unit IV:
IV
9.00
Reimann Integration , Lower and upper Reimann integrals, Properties of Reimann integration , Mean value theorem of Integral calculus , Fundamental theorem of integral calculus.
Unit V:
V
9.00
Uniform convergence , Sequence and series of function – pointwise and uniform convergence , Weirstrass M- Test, Abel’s and Drichlet’s Test for uniform convergence of series of functions . Continuity of the sum functions of the limit fuctions.
Essential Readings:
- Shanti Narayan, A course of Mathematical Analysis ,S.Chand and Co. New Delhi, 1995.
- T.M.Apostol , Mathematical Analysis , Norosa Publishing House, New Delhi, 2000.
- K.C.Sarangi , Real Analysis and Metric spaces, Ramesh Book Depot, Jaipur, 2006.
References:
- Jain and Kaushik, An introduction to Real Analysis , S.Chand and Co., New Delhi, 1990.
- Undergraduate Analysis , S.Lang , Springer-Verlag, 1997.
- Real Analysis , R.R.Goldberg, Oxford and IBH publishing Company, New Delhi, 1999.
Academic Year:
