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 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
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
11.08 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.