## 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

- Complex number system.

Basic definitions and properties; topology of the complex plane; connectedness, domains. Riemann sphere, stereographic projection. - Differentiability.

Basic definitions and properties; Cauchy-Riemann equations, analytic functions. - Elementary functions.

Fundamental algebraic, analytic, and geometric properties. Basic conformal mappings. - 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. - Sequences and series.

Uniform convergence; power series, radius of convergence; Taylor series. - The local theory.

Zeros, the identity theorem, Liouville's theorem, etc. Maximum modulus theorem, Schwarz's Lemma. - Laurent series

Classification of isolated singular points; Riemann's theorem, the Casorati-Weierstrass theorem. - Residue theory.

The residue theorem, evaluation of certain improper real integrals; argument principle, Rouche's theorem, the local mapping theorem. - The global theory.

Winding number, general Cauchy theorem and integral formula; simply connected domains. - Uniform convergence on compacta.

Ascoli-Arzela theorem, normal families, theorems of Montel and Hurwitz, the Riemann mapping theorem. - Infinite products. Weierstrass factorization theorem.
- Runge's theorem. Applications.
- 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.