MAT519
Complex Variables I
Spring 2010
Instructor: Dr. IULIANA OPREA
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