MTH305: Complex analysis

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• 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, Möbius 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.

Recommended Reading

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