1. Real Analysis

Elementary Set Theory, Countable and uncountable sets, real number system as a complete ordered field, Archimedean property, supremum and infimum, sequences and series, convergence and absolute convergence. Various types of convergences, limit superior and limit inferior. Bolzano-Weierstrass theorem, Heine-Borel theorem. Continuity and uniform continuity, differentiability, sequences and series of functions and their convergence viz.

Cauchy’s Criteria and Weierstrass M-test for convergences. Riemann integration of
bounded functions and its various properties, monotone functions. Types of discontinuities,
Lebesgue measure, Lebesgue integral. Functions of several variables, directional
derivatives, partial derivative. Total derivative, mean value theorem for differentiable
functions.

2. COMPLEX ANALYSIS

Algebra of Complex numbers, complex planes, functions of a complex variable.
Continuity, differentiability, CR-equations, analytic functions. Necessary and sufficient
condition for analyticity, harmonic functions, harmonic Conjugate, contour integration.
Cauchy integral theorem, Cauchy’s Integral Formula, Liouville’s theorem, maximum
modulus principle, fundamental Theorem of algebra. Morera’s theorem, Schwarz lemma,
open mapping theorem, Taylor series expansion. Elementary Linear Transformations, Mobius transformation, conformal mapping, singularities and its types, Riemann’s theorem
on removable singularities, Laurent’s series expansion, Hurwitz theorem, Cauchy residue
theorem, integrals of rational and trigonometric functions by residue theorem.

3. TOPOLOGY

Definition of a metric space, Examples, Open and closed sets, compactness in metric
spaces, continuity, uniform continuity, complete metric spaces, Topological spaces,
Definition and examples, Elementary Properties, Interior, exterior and boundary of a set, closure of a set, Kuratowski Axioms, Neighbourhood system, Local base and base,
subspaces and relative topology, continuous maps, open maps, closed maps and their
characterization, Homeomorphism, Separation Axioms, Lebesgue Covering lemma. Product topology, weak topology, compactness and connectedness in topological spaces, Tychnoff’s theorem.