Syllabus Math 542

Math 542. Complex Variables I Instructor Syllabus

Textbooks used in past semesters:
Functions of One Complex Variable (2nd Edition),
Conway, Springer-Verlag, NY

An Introduction to Complex Function Theory (1st Edition)
Palka, 1991, Springer

1. Complex number system.
   Basic definitions and properties; topology of the complex plane; connectedness, domains. Riemann sphere, stereographic projection.
2. Differentiability.
   Basic definitions and properties; Cauchy-Riemann equations, analytic functions.
3. Elementary functions.
   Fundamental algebraic, analytic, and geometric properties. Basic conformal mappings.
4. Contour integration.
   Basic definitions and properties; the local Cauchy theory, the Cauchy integral theorem and integral formula for a disk; integrals of Cauchy type; consequences.
5. Sequences and series.
   Uniform convergence; power series, radius of convergence; Taylor series.
6. The local theory.
   Zeros, the identity theorem, Liouville's theorem, etc. Maximum modulus theorem, Schwarz's Lemma.
7. Laurent series
   Classification of isolated singular points; Riemann's theorem, the Casorati-Weierstrass theorem.
8. Residue theory.
   The residue theorem, evaluation of certain improper real integrals; argument principle, Rouche's theorem, the local mapping theorem.
9. The global theory.
   Winding number, general Cauchy theorem and integral formula; simply connected domains.
10. Uniform convergence on compacta.
    Ascoli-Arzela theorem, normal families, theorems of Montel and Hurwitz, the Riemann mapping theorem.
11. Infinite products. Weierstrass factorization theorem.
12. Runge's theorem. Applications.
13. Harmonic functions.
    Definition and basic properties; Laplace's equation; analytic completion on a simply connected region; the Dirichlet problem for the disk; Poisson integral formula.