Unit 1

Elementary properties and examples of analytic functions, Power series, Analytic function, Riemann Stieltjes integrals.(Chapter 3 Sections 1, 2 and Chapter 4 Section 1 of Text)

Unit 2

Power series representation of an analytic function, Zeros of an analytic function, Liouville’s Theorem, Maximum Modulus Theorem, Index of a closed curve.(Chapter 4 – Sections 2, 3 and 4 of Text)

Unit 3

Cauchy’s Theorem and integral formula, Homotopic version of Cauchy’s Theorem, Simple connectivity, Counting zeros: The open Mapping Theorem, Goursat’s Theorem. (Chapter 4 Sections 5, 6, 7 and 8 of Text)

Unit 4

Singularities: Classification, Removable, Pole and Essential Singularity, Laurent Series, Casorati Weierstrass Theorem, Residue theorem, The argument principle, Rouche’s Theorem. (Chapter 5 Sections 1, 2, and 3 of Text)

Unit 5

The extended plane and its spherical representation, Analytic function as mapping, Mobius transformations, The maximum principle, Schwarz’s Lemma.(Chapter 1 Section 6, Chapter 3 Section 3, Chapter 6 Section 1 and 2 of Text)

CO1: To understand the basic idea of analytic functions, power series etc.
CO2: Ability to understand power series representation of Analytic function and zero’s of analytic functions
CO3: Understand Cauchy’s Theorem and integral formula, Homotopic version of Cauchy’s Theorem
CO4: To understand Singularities and Residue theorem
CO5: The extended plane and its spherical representation, Analytic function as mapping, Mobius transformations

Textbooks

1. John B Conway, Functions of One Complex Variable, Springer.

References

1. Elias Stein and Rami Shakarchi, Complex Analysis, New Age Publishers.
2. Lars V Ahlfors, Complex Analysis,Tata McGraw-Hill.