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
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.
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.
Finish Applications to real integrals
Applications to real integrals
Finish the Argument Principle
Change in function arguments; the Argument Principle II; Rouché's Theorem.
Begin the Argument Principle
Multiplicites of zeros; the Argument Principle I.
Finish The Residue Theorem
Begin The Residue Theorem
Definitions and examples of singularities; Casorati-Weierstrass Theorem; singularities and limits; meromorphic functions.
Finish Laurent series
Representations of holomorphic functions on annuli as Laurent series; examples.
Consequences of Cauchy's Integral Formulas
Morera's Theorem; holomorphic functions are smooth; Liouville's Theorem; the Fundamental Theorem of Algebra; holomorphic implies analytic.
Cauchy's Integral Formula
Cauchy's Integral formula; "higher" Cauchy Integral Formulas; examples.
Finish Cauchy's Theorem
No class. Take-home Exam 1 assigned, due Monday at class time.
"Triangular" Cauchy's Theorem; the general Cauchy's Theorem.
Finish Complex integration, part 2.
Fall break. No class.
Complex integration, part 2
The Fundamental Theorem of Calculus for complex integrals; the "triangular" complex antiderivative theorem.
Complex integration, part 1
Smooth curves in the complex plane; the integral of a complex-valued function of a complex variable.
Finish Power series, part 3
Continue Power series, part 3
Power series, part 3
The complex exponential, sine, and cosine functions; logarithm functions; the Riemann surface of the logarithm function; complex powers.
Power series, part 2
Analytic functions; analytic functions are holomorphic.
Power series, part 1
Radii of convergence; circles of convergence; funny behavior on the circle of convergence.
Finish Limits and differentiability, part 3
Limits and differentiability, part 3
More on the CR equations and real versus complex differentiability; orientations; holomorphic functions; "constancy" theorems.
Limits and differentiability, part 2
Real differentiability versus complex differentiability; the Cauchy-Riemann equations.
Finish Limits and differentiability, part 1
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.
Finish Prelude to complex analysis, part 3
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.
Prelude to complex analysis, part 2
Closures and interiors. Intersections and unions of open and closed sets.
Prelude to complex analysis, part 1
Completeness of the complex numbers. Neighborhoods, open sets, closed sets.
Basic properties of complex numbers, part 2
The metric structure on the complex plane. Arguments and polar forms. Roots, including roots of unity.
Basic properties of complex numbers, part 1
Definition of complex numbers. Algebraic operations, conjugates, and moduli. The complex plane. Triangle inequalities.
Real analysis primer, part 2
A review of the derivative and the Fundamental Theorems of Calculus.
Real analysis primer, part 1
A review of sequences, series, limits, continuity.