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

Web page:  https://faculty.math.illinois.edu/~fboca/welcome.html

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

Prerequisites:  MATH 446 and MATH 447, or MATH 448. You may contact the instructor for any queries or concerns.

Topics will include:

• 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:  Tuesday: 5:15-6:15 pm, Thursday: 5-6 pm, or by appointment.

Grading policy: Comprehensive final exam: 45%; Two midterm exams: 2x20 = 40%; Homework: 15%.

Homework assignments:

HW # 1 (due Friday Sep 5): Sec.I.1: 2,5,13,16,19; Sec.I.2: 1,8,11,17,19.

HW # 2 (due Friday Sep 12): Sec.I.3: 2,7,11; Sec.I.4: 2,4; Sec.I.5:: 5,9,11,12,15.

HW # 3 (due Wednesday Sep 24): Sec.I.5: 7; Sec.II.1: 4,6,8; Sec.II.2: 3,17; Sec.II.3: 1,2,6,7.

HW # 4 (due Monday Oct 6): Sec.II.3: 8,12; Sec.III.1: 4,7; Sec.III.2: 13,15; Sec.III.3: 10,16; Sec.III.4: 8,9.

HW # 5 (due Friday Oct 17): Sec.III.5: 3,4,5; Sec.III.7: 9,11,12,13,14,15,16.

HW #6 (due Wednesday Oct 29)

HW #7 (due Monday Nov 10)

HW #8 (due Friday Nov 21)

HW #9 (due Wednesday Dec 3)

HW #10 (due Wednesday Dec 10)

Midterm exams: Midterm 1: Mon Oct 6, 5-7 pm (room: 441 Altgeld Hall); Midterm 2: Th Dec 4, 5-7 pm (room: 443 Altgeld Hall).

Final exam: 1:30-4:30 pm, Wed Dec 17, 441 Altgeld Hall