Syllabus: Real and complex number systems. Limits of sequences. Monotonic sequences. Limits superior and limits inferior. Convergence of a series. Absolute and conditional convergence. Power series over real and complex numbers and their radius of convergence. Bolzano-Weierstrass Theorem, Cantor and Heine-Borel Theorems. Point wise and uniform continuity. Sequences and series of functions. Point wise and uniform convergence of sequence of functions. Integrals and derivatives of sequences and series of functions. Elementary transcendental functions. Improper integrals, Riemann-Stieltjes integral. Idea of Lebesgue integral, Weierstrass approximation Theorem, Inverse function theorem. Implicit function theorem.
Desirable: Multivariable Calculus.
- T. M. Apostol: Mathematical Analysis, Second Edition, Addison-Wesley Publishing Company, 1974.
- T. Tao: Analysis I, Hindustan Book Agency, 2017.
- T. Tao: Analysis II, Hindustan Book Agency, 2017.
- K. A. Ross: Elementary analysis: the theory of calculus, Springer, 2nd edition, 2013.