Math 448 Complex Variables
Instructor Syllabus

Textbook: S.D. Fisher, Complex Variables, Dover, 1999.

Chapter 1: The Complex Plane (8 hours)
1.1. Complex numbers (2 hours)
1.2-1.3. Geometry of the complex plane (1)
1.4. Functions and limits (1)
1.5. Exponential, logarithm, and trigonometric functions (2)
1.6. Line integrals and Green’s theorem (2) (can cover 1.6 between 2.2 and 2.3)

Chapter 2: Basic Properties of Analytic Functions (16)
2.1 Analytic and harmonic functions; Cauchy-Riemann equations (2)
2.2. Power series (1)
2.3- 2.3.1. Cauchy’s theorem and Cauchy’s formula (3)
2.4. Consequences of Cauchy’s formula (3)
2.5. Isolated singularities (4)
2.6. Residue theorem and evaluation of definite integrals (3)

Chapter 3: Analytic Functions as Mappings (14)
3.1. Zeros of an analytic function (3)
3.2. Maximum modulus and mean value (3)
3.3. Linear fractional transformation (2)
3.4. Conformal mapping (3)
3.5. Riemann mapping theorem (with sketch of the proof) and Schwarz-Christoffel transformations (optional) (3)  

Exams and Leeway: 5 hours

Total:  43 hours

Updated: UAC 4/12/16