Math 650
Several Complex Variables

Harold P. Boas

Fall 2011

Lecture notes (as of November 30)

Course description

This course is an introduction to the theory of functions of several complex variables. The emphasis is on the part of the theory that intersects with analysis and with partial differential equations.
Here are some of the topics that will be discussed (although not in linear order).

• Multi-variable power series
  • Reinhardt domains
  • domains of convergence
  • the Hartogs phenomenon
  • entire functions
• Integral representations
  • Cauchy integral
  • Bochner–Martinelli integral
  • Bergman kernel function
• Notions of convexity
  • linear convexity
  • polynomial convexity
  • holomorphic convexity
  • pseudoconvexity
• Levi problem
• d-bar problem
• Holomorphic mappings

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.

Venue

The course meets 11:10–12:25 on Tuesday and Thursday in room 624 of the Blocker building.

Course website

http://www.math.tamu.edu/~boas/courses/650-2011c/

Textbook

There is no required textbook. I have asked the library to put 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.
• Volker Scheidemann, Introduction to complex analysis in several variables, Birkhäuser, 2005; QA331.7 .S34 2005.

Grading

Grades will be based on class participation and homework.

Office hours

During the Fall 2011 semester, my office hour in Milner 202 is 2:00–3:00 in the afternoon on Monday and Wednesday; 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.

TAMU policy statements

Americans with Disabilities Act (ADA)

This federal anti-discrimination statute provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services in Room B118 of Cain Hall, or call 979-845-1637. For additional information visit the website of Disability Services.

Academic integrity

The Aggie Honor Code states: “An Aggie does not lie, cheat or steal, or tolerate those who do.” Information about the Honor Council Rules and Procedures can be found at the website of the Aggie Honor System Office.

Attendance

See Student Rule 7.