Syllabus for
Math 448. Complex Variables
Textbook: S.D. Fisher, Complex Variables, Dover, 1999.
- Complex numbers and functions. (4)
- Analytic functions: Cauchy-Riemann equations, exponential, trigonometric, logarithm and power functions, the Riemann surface for log z, harmonic functions. (6)
- 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)
- 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)
- Conformal mapping, bilinear transformations, inverse mappings and univalent functions, global mapping theorems, the Riemann mapping theorem. (6)
- Uniform convergence for sequences and series, power series. (4)
(Numbers of hours per topic are in parentheses. Total = 40 hours.)