MAT 442 Complex Analysis

**Catalog description:** Complex functions, derivatives and integrals; analytic functions and Cauchy's Integral Theorem; power series and Laurent series; residue theory and its applications to real integration; uniform convergence of a sequence of analytic functions; conformal mapping.

**Time & Place:** 10:20-11:15AM MWF, Shineman 194 **Syllabus:** link**Office:** MCC 175**Office hours:** 12:45-1:45PM MWF, 12:15-1:15PM Tu, and by appointment

Week 15

No homework

12.3 Fri

Finish Conformal maps and MĂ¶bius transformations

Three main types of MĂ¶bius transformations that generate all; circles are preserved by MĂ¶bius transformations; mapping three points to three points.

12.1 Wed

Begin Conformal maps and MĂ¶bius transformations

Definition of conformal maps; conformal maps are angle-preserving; definition of MĂ¶bius transformations, and closure under inverses and compositions.

11.29 Mon

Finish Applications to real integrals

Week 14

No homework

Week 13

No homework

11.19 Fri

Exam 2

11.17 Wed

Finish the Argument Principle

Change in function arguments; the Argument Principle II; RouchĂ©'s Theorem.

11.15 Mon

Begin the Argument Principle

Multiplicites of zeros; the Argument Principle I.

11.12 Fri

Finish The Residue Theorem

11.08 Mon

Continue Singularities

11.05 Fri

Singularities

Definitions and examples of singularities; Casorati-Weierstrass Theorem; singularities and limits; meromorphic functions.

11.03 Wed

Finish Laurent series

11.01 Mon

Laurent series

Representations of holomorphic functions on annuli as Laurent series; examples.

10.29 Fri

Consequences of Cauchy's Integral Formulas

Morera's Theorem; holomorphic functions are smooth; Liouville's Theorem; the Fundamental Theorem of Algebra; holomorphic implies analytic.

10.27 Wed

Cauchy's Integral Formula

Cauchy's Integral formula; "higher" Cauchy Integral Formulas; examples.

10.25 Mon

Finish Cauchy's Theorem

Week 9

No homework

10.22 Fri

No class. Take-home Exam 1 assigned, due Monday at class time.

10.18 Mon

Finish Complex integration, part 2.

10.15 Fri

Fall break. No class.

10.13 Wed

Complex integration, part 2

The Fundamental Theorem of Calculus for complex integrals; the "triangular" complex antiderivative theorem.

10.11 Mon

Complex integration, part 1

Smooth curves in the complex plane; the integral of a complex-valued function of a complex variable.

10.08 Fri

Finish Power series, part 3

10.06 Wed

Continue Power series, part 3

10.04 Mon

Class cancelled

10.01 Fri

Power series, part 3

The complex exponential, sine, and cosine functions; logarithm functions; the Riemann surface of the logarithm function; complex powers.

09.27 Mon

Power series, part 1

Radii of convergence; circles of convergence; funny behavior on the circle of convergence.

09.24 Fri

Finish Limits and differentiability, part 3

09.22 Wed

Limits and differentiability, part 3

More on the CR equations and real versus complex differentiability; orientations; holomorphic functions; "constancy" theorems.

09.20 Mon

Limits and differentiability, part 2

Real differentiability versus complex differentiability; the Cauchy-Riemann equations.

09.17 Fri

Finish Limits and differentiability, part 1

09.15 Wed

Begin Limits and differentiability, part 1

Limits and continuity of complex-valued functions. The action of degree-1 complex polynomials. Derivatives of complex-valued functions and intuition.

09.13 Mon

Finish Prelude to complex analysis, part 3

09.10 Fri

Prelude to complex analysis, part 3

Complex-valued functions. The squaring function as a branched double cover. The action of power functions on the unit circle.

09.03 Fri

Prelude to complex analysis, part 2

Closures and interiors. Intersections and unions of open and closed sets.

09.01 Wed

Prelude to complex analysis, part 1

Completeness of the complex numbers. Neighborhoods, open sets, closed sets.

08.30 Mon

Basic properties of complex numbers, part 2

The metric structure on the complex plane. Arguments and polar forms. Roots, including roots of unity.

08.27 Fri

Basic properties of complex numbers, part 1

Definition of complex numbers. Algebraic operations, conjugates, and moduli. The complex plane. Triangle inequalities.

08.25 Wed

Real analysis primer, part 2

A review of the derivative and the Fundamental Theorems of Calculus.