MAT519
Complex Variables I

Spring 2010

Instructor: Dr. IULIANA OPREASpring 2010

http://www.math.colostate.edu/~juliana

Office & Phone #: Weber 123, 491-6751;

Time/Location: MWF 3:00 - 3:50PM, Engineering E206

Textbook:

Complex Variables, Introduction and Applications, M.J. Ablowitz and A.S. Fokas, Cambridge University Press, 2nd ed. 2003.

Supplemental Material

1. Functions of a Complex Variable, 2005 (SIAM publishing), by George Carrier, Max Krook, and Carl E. Pearson (applied text),

2. Applied Complex Analysis, N.H. Asmar, Prentice Hall 2002

3. Function Theory of One Complex Variable (AMS Publishing), by Robert Greene and Steven Krantz

4. Analytic Function Theory, vols. 1 and 2 (Chelsea Publishing), by Einar Hille (a classic with a wealth of information),

5. Theory of functions of a complex variable (Chelsea Publishing) by Alexsei I. Markushevich (1180pp! and a great reference to have around)

Overview: This course is an introduction to the theory of complex valued functions, with equal emphasis given to the rigorous mathematical development of the subject as to practical applications of the theory.

Syllabus:

(1) Complex numbers and elementary functions (complex numbers, properties, stereographic projection, elementary functions, limits, continuity, differentiation)

(2) Analytic functions and integration (Cauchy-Riemann equations, multivalued functions, Riemann surfaces, integration, Cauchy's theorem, Cauchy's formula, generalizations, Liouville's theorem, Morera's theorem)

(3) Sequences, series and singularities (definitions, Taylor series, Laurent series, singularities, analytic continuation, infinite products)

(4) Residue calculus and applications (The residue theorem, definite integrals, principal-value integrals, integrals with branch points, the argument principle, Fourier and Laplace transforms)

(5) As time permits, a special topic chosen to suit class interest (possibilities: conformal mappings, asymptotic evaluation of integrals, elliptic functions, the prime number theorem, etc)

Homework (55%): 10 assignments, collected on Friday. Late homework is not accepted

In class Midterm (15%): March, TBA

Final Exam (30%): TBA