### MATH 5120: Complex Function Theory I

**Description:** Complex plane, Riemann sphere, Euler's formula, complex differentiable functions and Cauchy-Riemann equations, conformal maps, linear fractional transformations. Integration along simple rectifiable curves, Cauchy-Goursat and Morera's theorems, Cauchy integral formula, Cauchy estimates and Schwarz lemma. Power series and the disk of convergence, Taylor and Laurent series, classification of singularities. The argument principle, winding numbers and Rouche's theorem. Cauchy's residue theorem and its use in evaluating real-valued integrals. Maximum modulus, Liouville and Picard theorems, the Fundamental Theorem of Algebra, Schwarz reflection principle. Harmonic functions and harmonic conjugates. Normal families and Montel's theorem. The Riemann mapping theorem. A practical purpose of the class is to prepare students to take the qualifying exams.

For prelim preparation, see the prelim study guide.

**Prerequisites:** MATH 5110.

**Credits:** 3

**Sections: **Spring 2009 on Storrs Campus

Course | Sec | Comp | Time | Room | Instructor |
---|---|---|---|---|---|

5120 | 1 | Lecture | TuTh 2:00-3:15 | MSB411 | Rogers, Luke |