M551 - Spring 2006

Math 551 - Spring 2006

TEXTBOOK: Basic Complex Analysis, 3rd ed., J. Marsden and M. Hoffman
ALTERNATE READING
Complex Variables (Schaum's Outline), M. Spiegel
Complex Variables, Levinson and Redheffer
INSTRUCTOR: Mark Pernarowski
OFFICE: Wilson 2-236
OFFICE HOURS: Schedule
PHONE: 994-5356
GRADING SYSTEM:

Topics Covered:

Basic Properties of Analytic Functions

Cauchy-Riemann equations

Elementary functions, branches and branch cuts

Elementary Topology, connected sets

Analytic functions and Harmonic functions

Inverse function Theorem

Cauchy's Theorem

Contour integrals, Path Independence

Cauchy's Theorem

Cauchy's integral formula (for the function and derivatives)

Liouville's Theorem, Fundamental Theorem of Algebra

Morera's Theorem

Mean Value Theorems

Local and Global Maximum Moduli Theorems

Analytic Series

Uniform Convergence: Integration and differentiation of series

Power series, Binomial Series and Taylor's Theorem

Analytic continuation, Monodromy Theorem

poles, essential singularities

Laurent series

Zeros of analytic functions

Casorati-Weierstrass and Picard Theorems

Calculus of Residues and Integrals

Residue Theorem

Partial Fraction Theorem

Jordan's Lemma and Integral Bounding

Definite integrals via residues

Definite integrals about cuts

Principal Value integrals

Summation of Series

Conformal Mappings

Mapping properties of elementary functions

Fractional linear transformations: inverse points, geometrical properties

Riemann Mapping Theorem (no proof)

Solution of Dirchlet BVP via conformal mapping

Homework Assignments