Complex Variable Theory
Revised: September, 2014
Course Description
The complex number system, limits, continuity, derivatives, transcendental and multiple
valued functions, integration. Prerequisite: MATH 256. Three semester hours.
Student Learning Objectives
By the end of the course students will be able to:
- Represent complex numbers algebraically and geometrically;
- Analyze complex functions both algebraically and geometrically;
- Define and analyze limits and continuity for complex functions as well as consequences
of continuity;
- Use the Cauchy-Riemann equations to analyze analytic functions;
- Analyze sequences and series of analytic functions and types of convergence;
- Evaluate complex contour integrals directly and by the fundamental theorem, apply
the Cauchy integral theorem in its various versions, and the Cauchy integral formula;
and
- Represent functions as Taylor, power and Laurent series, classify singularities and
poles, find residues and evaluate complex integrals using the residue theorem.
Text
Zill and Shanahan,
A First Course in Complex Analysis with Applications, Jones and Bartlett, 2003.
Grading Procedure
Grading procedures and factors influencing course grade are left to the discretion
of individual instructors, subject to general university policy.
Attendance Policy
Attendance policy is left to the discretion of individual instructors, subject to
general university policy.
Course Outline
-
Chapter 1: Complex Numbers and the Complex Plane (8 days)
Complex numbers and their properties; complex plane; polar form of complex numbers;
powers and roots; sets of points in the complex plane.
-
Chapter 2: Complex Functions and Mappings (8 days)
Complex functions; complex functions as mappings; linear mappings; special power
functions; reciprocal function; limits and continuity.
-
Chapter 3: Analytic Functions (4 days)
Differentiability and analyticity; Cauchy-Riemann equations; harmonic functions.
-
Chapter 4: Elementary Functions (4 days)
Exponential and logarithmic functions; complex powers; trigonometric and hyperbolic
functions.
-
Chapter 5: Integration in the Complex Plane (7 days)
Real integrals; complex integrals; Cauchy-Goursat Theorem; Independence of Path;
Cauchy’s Integral Formulas and Their Consequences.
-
Chapter 6: Series and Residues (7 days)
Sequences and series; Taylor series; Laurent Series; Zeros and Poles; Residues and
Residue Theorem.
Times above include review and testing.
