Math 542: Complex Variables I

Fall 2008

Instructor: Florin Boca

Office: 359 Altgeld Hall

Phone: 244-9928

E-mail: fboca at math dot uiuc dot edu

Textbook: Complex Analysis, E. Freitag and R. Busam, Springer (Universitext), 2005.

Prerequisites: MATH 446 and 447, or MATH 448.

Topics:

  • 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.

    • Lectures: MWF 12:00-12:50, 441 Altgeld Hall

    Office hours: TBD.

    Last modified: April 14, 2008