**Duration:** 21 weeks.

**Number of lectures per week:** 3

**Assessment:** Regular assignments.

**End-of-year Examination:** One 3-hour examination

**Description: **

- Review of the definition of analytic functions, the Cauchy-Riemann
equations, differentiation of power series, and contour integrals.
Various forms of Cauchy's theorem and the Cauchy integral formula;
winding numbers and homotopy.
- Logarithms,
simple connectedness and antiderivatives.
- Identity theorem for analytic functions, maximum modulus theorem.
- Open mapping theorem, argument principle, inverses of
analytic functions and Rouché's theorem. Removable
singularities, Casorati-Weierstrass theorem, Residue
theorem.
- Metric space structures on H(G) and C(G).
Boundedness and compactness in H(G).
- Normal families; metric space structure of M(G).
- Continuous linear operators and dual spaces.
Hahn-Banach theorem (without proof) and applications.
Runge's theorem.
- Hurwitz's theorem, the Schwarz lemma, the Riemann mapping theorem.

**Textbooks:**

- [1]
L. V. Ahlfors,
*Complex Analysis,*Third Edition, McGraw-Hill, New York, 1978. - [2]
John B. Conway,
*Functions of One Complex Variable,*Second Edition, Graduate Texts in Mathematics 11, Springer-Verlag, New York, 1978. - [3]
Reinhold Remmert,
*Theory of Complex Functions,*Graduate Texts in Mathematics 122, Springer-Verlag, New York, 1991. - [4]
W. Rudin,
*Real and Complex Analysis,*Second Edition, McGraw-Hill, New York, 1974.