 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
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 Möbius transformations

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

12.1 Wed

Begin Conformal maps and Möbius transformations

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

11.29 Mon

Finish Applications to real integrals

Week 14

No homework

11.22 Mon

Applications to real integrals

Week 13

No homework

11.19 Fri

Exam 2

11.17 Wed

Finish the Argument Principle

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

11.15 Mon

Begin the Argument Principle

Multiplicites of zeros; the Argument Principle I.

11.12 Fri

Finish The Residue Theorem

11.10 Wed

Finish Singularities

Begin 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.

10.20 Wed

Cauchy's Theorem

"Triangular" Cauchy's Theorem; the general Cauchy's Theorem.

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.

09.29 Wed

Power series, part 2

Analytic functions; analytic functions are holomorphic.

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.

08.23 Mon

Real analysis primer, part 1

A review of sequences, series, limits, continuity.