Complex Analysis I (MAT 542)

1. The field of complex numbers, geometric representation of complex numbers

2. Analytic functions
   • Definition, Cauchy-Riemann equations
   • Elementary theory of power series, uniform convergence
   • Elementary functions: rational, exponential and trigonometric functions
   • The logarithm

3. Analytic functions as mappings
   • Conformality
   • Linear fractional transformations
   • Elementary conformal mappings

4. Complex integration
   • Line integrals and Cauchy's theorem for disk and rectangle
   • Cauchy's integral formula
   • Cauchy's inequalities
   • Morera's theorem, Liouville's theorem and fundamental theorem of algebra
   • The general form of Cauchy's theorem

5. Local properties of analytic functions
   • Removable singularities, Taylor's theorem
   • Zeros and poles, classification of isolated singularities
   • The local mapping theorem
   • The maximum modulus principle, Schwarz's lemma

6. The calculus of residues
   • The residue theorem
   • The argument principle
   • Rouche's theorem
   • Evaluation of definite integrals

7. Power series
   • Weierstrass theorem
   • The Taylor and Laurent series
   • Partial fractions and infinite products
   • Normal families

8. The Riemann mapping theorem

9. Harmonic functions
   • The mean-value property
   • Harnack's inequality
   • The Dirichlet problem

Typical references:

• Lars V. Ahlfors, Complex analysis: an introduction to the theory of analytic functions of one complex variable,
  3^rd ed.; McGraw-Hill, 1979;
• John B. Conway, Functions of one complex variable,
  2^nd ed.; Springer-Verlag, 1978.