Math 448. Complex Variables
Syllabus for Instructors

1. Complex numbers and functions. (4)

2. Analytic functions: Cauchy-Riemann equations, exponential, trigonometric, logarithm and power functions, the Riemann surface for log z, harmonic functions. (6)

3. Complex integration: contour integration, the Cauchy integral theorem and Cauchy- Goursat theorem for star-shaped regions, the Cauchy integral formula, Taylor's series, uniqueness, the maximum principle, isolated singularities, Laurent series. (13)

4. Residue theory: Simply connected domains, the residue theorem, integrals over the real axis, improper integrals and principal values, integrands with branch points, principle of the argument, Rouché's theorem. (7)

5. Conformal mapping, bilinear transformations, inverse mappings and univalent functions, global mapping theorems, the Riemann mapping theorem. (6)

6. Uniform convergence for sequences and series, power series. (4)

(Numbers of hours per topic are in parentheses. Total = 40 hours.)

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Syllabus Revised 6/21/99