Complex Analysis-I

Syllabus Details

• Unit I: Complex numbers, holomorphic functions, and branches of logarithm: basic properties of complex numbers; holomorphic functions; Cauchy-Riemann equations; complex exponential function; branches of logarithm.
• Unit II: Power series, and Mobius maps: complex power series and radius of convergence; Mobius maps on the Riemann sphere, and their properties; biholomorphic maps.
• Unit III: Complex integration along a path, Cauchy’s integral formula, and power series expansion of holomorphic functions.
• Unit IV: Fundamental theorems about holomorphic functions: Liouville’s theorem; zeros of holomorphic functions; Maximum modulus principle; Open mapping theorem; Weierstass’ uniform convergence theorem.
• Unit V: Winding number, and singularities: winding number; counting zeros in a region bounded by a smooth path; three types of singularities; Casorati-Weierstrass theorem; Laurent series.
• Unit VI: A few advanced topics in Complex Analysis: Residue theorem; Argument principle - counting zeros and poles of a meromorphic function; Rouche’s theorem; Hurwitz theorem; Schwarz lemma; a brief mention of harmonic functions.

Course Learning Objective

1. Identify holomorphic functions, explain branches of logarithm, compute complex integrals, and solve elementary problems.
2. Explain Cauchy’s theory, power series expansion, and fundamental theorems about holomorphic functions, and solve elementary problems about them.
3. Identify types of singularities, compute residues, explain properties of harmonic functions and solve elementary problems about them.
4. Analyze the proofs of results in Complex Analysis with a critical mind and use those results and the proof-techniques to analyze and decide, with suitable justification, whether a given statement in Complex Analysis is true or false
5. Interpret the theory and theoretical structure of Complex Analysis as a prerequisite for the future study subjects such as Operator Theory, Riemann Surfaces, Analytic Number Theory, and Complex Algebraic Geometry.

Recommended Books

1. Ahlfors, Lars V., Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, third edition. International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978.
2. Churchill, Ruel V. and Brown, James Ward, Complex Variables and Applications, fourth edition, McGraw-Hill Book Co., New York, 1984.
3. Conway, John B., Functions of One Complex Variable, II, Graduate Texts in Mathematics, 159, Springer-Verlag, New York, 1995.
4. Narasimhan, Raghavan and Nievergelt, Yves, Complex Analysis in One Variable, second edition, Birkhauser Boston, Inc., MA, 2001.