University of Rhode Island      Department of Mathematics

MTH 462  Complex Variables

TuTh 2-3:15pm Lippitt 205

Instructor  Dr. Mark Comerford
Office  Lippitt 102 F
Phone  874 5984
Office Hours
 Wed 2 - 4 pm
  or by appointment
Text  Stephen D. Fisher
Complex Variables
(Second Edition),
 ISBN 0-486-40679-2
Prerequisites  MTH 243 or equivalent

Lectures and Homework Problems

Section Problems
1.1 Complex Numbers and the Complex Plane 1f, 3d, 5d, 6f, 11, 13b
1.2 Some Geometry 2, 3, 5, 13, 15, 25, 28
1.3 Subsets of the Plane 1 - 8
1.4 Functions and Limits 1, 3, 5, 7, 11, 13, 15, 17, 21, 31, 34, 39
1.5 The Exponential, Logarithm and
      Trigonometric Functions
1, 4, 5, 6, 11, 13, 15, 19, 27
1.6 Line Integrals and
      Green's Theorem
1, 3, 5, 7, 8, 10, 11, 12
2.1 Analytic and Harmonic       Functions,
      Cauchy-Riemann Equations
1a, 1g, 3, 5, 7, 9, 14, 18, 20a, 20c , 21
2.2 Power Series 1- 5, 7 - 11, 15, 16, 22a, 22b
2.3 Cauchy's Theorem 1, 3, 4, 7, 10, 14, 15
2.4 Consequences of Cauchy's
1, 2, 4, 6, 9, 10, 12, 14, 17, 20
2.5 Isolated Singularities 1, 2, 5, 6, 7, 8, 10, 12, 16, 22a, 22b, 22c
2.6 The Residue Theorem and       Applications 1, 2, 4, 5, 9, 10, 16, 17, 20, 22
3.1 Zeroes of an Analytic Function
3.2 Maximum Modulus and Mean       Value
3.3 Linear Fractional       Transformations
3.4 Conformal Mappings



Midterm  2-3:15pm, Tuesday October 18  Lippitt 205
Final  TBA   Lippitt 205



Quizzes   20%
Projects   20%
Midterm   25%
Final   35%

Course Description

This course is an introduction to the wonderful world of complex variables and complex analysis. No initial knowledge of complex numbers is needed, however, some multivariable calculus is required and so Math 243 is a prerequisite. We will cover the material in the book until the residue theorem and its applications, especially its usefulness in doing certain types of real integrals (what physicists and engineers refer to as contour integration).

The emphasis will be mostly on developing the techniques of complex variables such as manipulating power series and working with integrals over paths in the complex plane. As much as possible, we will look at the visual side of the subject, as a working knowledge of elementary complex analysis can lead to many beautiful pictures, not least the fractal sets of complex dynamics such as the famous Mandelbrot set. With this in mind, there will be a project towards the end of the semseter, worth twenty percent of the overall grade. Some of these projects will be visual in nature and will involve using software such as Mathematica or Maple. Others will involve reading and presenting topics somewhat more advanced than those covered in class.

In addition to the project, there will be one midterm (provisional date Tuesday October 18), a final and bi-weekly quizzes. The material covered in the quizzes will be strongly based on the assigned homework with the expectation that the problems should be relatively straightforward for students who have mastered the last two week's homework problems.


Project 1: The Mandelbrot Set

Project 2: Hyperbolic Tessellations

Project 3: Newton's Method

Project 4: Escher's Art

Project 5: Complex Mappings and Spirograph Patterns

Project 6: The Riemann Mapping Theorem

Goals and Objectives

The goals of the course are to have you develop the basic skills presented in one semester of complex variables.

At the conclusion of this semester you should be able to:

1. Understand what an analytic function is and be able to test a function for analyticity.

2. Understand the properties of simple analytic functions such as polynomials, the exponential and the logarithm.

3. Derive and manipulate (e.g. multiply, compose, differentiate) Taylor series.

4. Carry out complex path integrals.

5. Understand and be able to apply Cauchy's theorem and Cauchy's integral formula, including the use of the winding number.

6. Understand the nature of the three different kinds of isolated singularity of an analytic function.

7. Develop and manipulate Laurent series.

8. Compute residues and use the residue theorem to calculate various types of integral.

Special Accommodations

Students who need special accomodations and who have documentation from Disability Services should make arrangements with me as soon as possible. Students should conact Disability Services for Students, Office of Student Life, 330 Memorial Union, 874-2098.