Math 542: Complex Variables I
Instructor: Florin Boca
Office: 359 Altgeld Hall
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
- 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
- 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
Grading policy: Comprehensive final exam: 45%; Two midterm
exams: 2x20 = 40%;
HW # 1 (due Friday Sep 5): Sec.I.1: 2,5,13,16,19;
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;
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;
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
Last modified: December 1, 2008