### MA 5360 Complex Analysis

#### Course Details

Unit I: Topology of the complex plane, Riemann sphere, limits, continuity and differentiability, Analytic functions, harmonic functions and multi-valued functions.

Unit II: Convergence of numerical series, Radius of convergence of power series, and power series as an analytic function, Laurent series.

Unit III: Cauchy's integral theorem, Cauchy integral formula, Morera's theorem, Taylor`s theorem, Laurent's theorem, Liouville's theorem, Schwarz lemma; Maximum Modulus Principle.

Unit IV: Conformal mappings, linear fractional transformations, Classification of singularities, Cauchy's residue theory and evaluation of real integrals.

#### Course References:

Text Books:

1. S. Ponnusamy and H. Silverman: Complex Variables with Applications, Birkhauser, Boston, 2006.

2. J.B. Conway: Functions of one Complex Variables, 2nd edition, Springer-Verlag, 1978.

Reference:

1. S. Ponnusamy: Foundations of Complex Analysis, Second Edition, Narosa Publishing House, 2005

2. L. Ahlfors: Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.

3. J.W. Brown and R.V. Churchil, Complex Variables and Applications, McGraw Hill, 2008

4. T.W. Gamelin: Complex Analysis, Springer-Verlag, 2001.