Math 424 Syllabus

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.

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