**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**

Last modified: *December 1, 2008*