Math 650
Several Complex Variables
Fall 2013

Course description
This course is an introduction to the theory of functions of several complex variables, emphasizing the part of the theory that intersects with analysis and with partial differential equations.
Here are some of the topics to be discussed.
  • multi-variable power series
    • Reinhardt domains
    • domains of convergence
    • the Hartogs phenomenon
    • entire functions
  • integral representations
    • the Cauchy integral
    • the Bochner–Martinelli integral
    • the Bergman kernel function
  • notions of convexity
    • linear convexity
    • polynomial convexity
    • holomorphic convexity
    • pseudoconvexity
  • the Levi problem
  • the ∂ problem
  • holomorphic mappings
Course objectives
By the end of the course, you should be able to
  • describe the similarities and the differences between one-dimensional function theory and multi-dimensional function theory;
  • explain the concept of domain of holomorphy;
  • read the research literature on multi-dimensional function theory.
Prerequisites
You should have some acquaintance at the first-year graduate level with both real analysis and (one-variable) complex analysis. The official prerequisites for this course are Math 608 and Math 618 (or equivalents).
Textbook
There is no required textbook. I have asked the campus library to put hard copies of the following books on reserve.
  • Lars Hörmander, An introduction to complex analysis in several variables, second edition, North-Holland, 1973; QA331 .H64 1973.
  • Steven G. Krantz, Function theory of several complex variables, second edition, American Mathematical Society, 2001; QA331.7 .K74 2001.
  • R. Michael Range, Holomorphic functions and integral representations in several complex variables, Springer-Verlag, 1986; QA331 .R355 1986.
Moreover, the indicated book by Range is available in electronic form from computers on campus. Another book that similarly is available electronically is Introduction to complex analysis in several variables by Volker Scheidemann (Birkhäuser, 2005; QA331.7 .S34 2005).
Meeting time and place
The course meets 11:10–12:25 on Tuesdays and Thursdays in room 624 of the Blocker building.
Grading
Grades will be based on class participation and homework. There will be no examinations (in particular, no final examination).
Course website
https://www.math.tamu.edu/~boas/courses/650-2013c/
Office hours
During the Fall 2013 semester, my office hour in Milner 202 is 2:00–3:00 in the afternoon on Tuesday, Wednesday, and Thursday; I am available also by appointment. The best way to contact me is via email to boas@tamu.edu. Telephone messages can be left at the Milner office of the Department of Mathematics, 979-845-7554.