Syllabus

Real Analysis

1. Lebesgue integration theory in one dimension, comparison with Riemann integral.
2. Measures, measurable functions, abstract integration theory, convergence theorems, construction of measures, decompositions of measures.
3. Absolute continuity of functions and measures, Radon-Nikodym Theorem, one-variable differentiation theory.
4. L ^p-spaces, Hilbert spaces, Hölder and Minkowski inequalities, Riesz Representation Theorem, density of continuous functions.
5. Product integration, Fubini's Theorem.
6. Metric spaces, compactness, completeness, separability, contraction mapping principle.

Complex Analysis

1. Complex numbers, polar representations, basic topology opf the complex plane and extended complex plane.
2. Elementary functions, polynomials, linear fractional transformations, exponential function, logarithm, trigonometric functions.
3. Holomorphic functions, Cauchy-Riemann equations, conformal mapping.
4. Contour integrals, Cauchy and Cauchy-Goursat Theorems, Cauchy's integral formula, Liouville's Theorem, Fundamental Theorem of Algebra.
5. Taylor and Laurent series.
6. Classification of singularities, residues, evaluation of real integrals, meromorphic functions, argument principle, Rouché's Theorem.
7. Maximum Modulus Theorem, Schwarz's Lemma

Examinations