Math 618 -- Syllabus

1. Review of basic complex analysis
2. Harmonic functions---Chapter X
3. Conformal mappings
   1. Review of Moebius transformations
   2. Schwarz-Christoffel Transformations
   3. Riemann Mapping Theorem (uniqueness part)
4. Normal Families---Chapter VII (Sections 2-4)
   1. Uniformly convergent sequences
   2. Definition of and properties of normal families
   3. Univalent (schlicht, simple, injective) analytic functions
   4. Riemann Mapping Theorem
   5. Normal families and spaces of continuous functions
5. Infinite products---Chapter VII (Sections 5-8)
   1. Weierstrass Factorization Theorem
   2. The Gamma Function
   3. The Riemann Zeta Function
   4. Connection with Wavelets
6. Runge's Theorem and Mittag-Leffler Expansions---Chapter VIII
7. Analytic continuation---Chapter IX (Sections 1-3)
8. Entire functions---Chapter XI and Chapter XII
   1. Classification
   2. Picard's Theorems
9. Applications of complex analysis