**[Cr:4, Lc:3, Tt:1, Lb:0]**

- The field of complex numbers, extended complex plane, convergence, subsets.
- Complex differentiation, analytic functions, polynomials, power series, exponential and trigonometric functions.
- Cauchy- Riemann equations, analytic functions as mappings, exponential function, logarithm, harmonic functions.
- Complex integration, Cauchy's theorem and integral formulas, power series representation.
- Zeros of analytic functions, index of a closed curve.
- Morera's theorem, Liouville's theorem, open mapping theorem. argument principle, Rouche's theorem.
- Poles and essential singularities, Casorati-Weierstrass theorem. residues, Laurent series.
- Maximum modulus principle, Schwarz lemma, Phragmen-Lindelof theorems.
- Conformality of analytic maps, MÃ¶bius transformations.

Additional Topics

- Gamma function, Riemann zeta function, prime number theorem.
- Analytic continuation, spaces of analytic functions and of meromorphic functions.
- Riemann mapping theorem, infinite products, Weierstrass factorization Theorem.

- Lars V. Ahlfors, Complex Analysis, McGraw-Hill (1979).
- John B. Conway, Functions of One Complex Variable, Springer (Graduate Texts in Mathematics Vol. 11) (1978).
- Theodore W. Gamelin, Complex Analysis, Springer (2003).
- Reinhold Remmert, Theory of Complex Functions, Springer, (Graduate Texts in Mathematics/Reading in Mathematics Vol. 122) (1998).
- Elias Stein and Rami Shakarchi, Complex Analysis, Princeton University Press (Princeton Lectures in Analysis) (2003).
- W. Tutschke and H. L. Vasudeva, An Introduction to Complex Analysis: Classical and Modern Approaches, Chapman & Hall/CRC (2005).