MTH 611 Complex Analysis
Spring 2012

Complex analysis, the study of functions of a complex variable, emerged in its modern form during the 19th century primarily from the seminal investigations by Cauchy, Riemann, and Weierstrass. Their distinct approaches -- Cauchy based his analysis on complex differentiability, Riemann studied mappings between domains in the complex plane, and Weierstrass relied on power series representations of functions -- are inextricably linked and have become to define the modern-day theory of analytic functions. Apart from its own intrinsic interest and beauty, complex analysis is studied as an indispensable tool in practically every major field of mathematics, and it has found applications in such varied subjects as computer graphics, cryptography, fluid mechanics, quantum theory, tomography, and so on.

MTH 611 Complex Analys will be taught as an introductory course to the theory of analytic functions. Topics covered will include:

• Analytic functions, Cauchy-Riemann equations
• Conformal maps, linear fractional transformations
• Cauchy theorems
• Maximum principle, open mapping theorem
• Argument principle
• Power series
• Singularities, Laurent series
• Residue theorem

We will use the book Ash, Novinger, Complex Variables, 2nd Ed., Dover 2007, as the course text. A pdf-version of the book can be downloaded gratis at http://www.math.uiuc.edu/~r-ash/CV.html.

The course is offered concurrently with MTH 619 Topics in Analysis, CRN 59554

Instructor: Juha Pohjanpelto
Office/Phone: Kidder Hall 368C, (541) 737-5156
Office hours: Wednesday noon–1:00 p.m., or by appointment
Email:
Homepage:

Prerequisites: Basics of differentiation and integration as covered e.g. in MTH 411.

 Learning objectives: Upon successfully completing this course a student • comprehends the notion of an analycity for complex functions and recognizes a large variety of analytic functions. • is capable of evaluating contour integrals involving complex functions and is familiar with the Cauchy integral theorems. • comprehends the maximum principle and recognizes its basic applications to the study of analytic functions. • appreciates Laurent expansions of analytic functions and is adept in applying the ensuing techniques to evaluating definite integrals. • is acquainted with the key applications of analytic functions.

Grading: Your course grade will be based on a mid-term exam, a final exam, and four homework assignments. The midterm exam counts 30%, the final exam 40%, and the homework assignments 30% towards the course grade.

• The midterm exam will be held on Friday, May 4.
• The homework assignments will be due on April 16, April 27, May 25, June 8.
• The due date for the final exam will be announced at a later time.

You can view the course calendar by clicking here