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Complex Variable Theory

Revised: September 2020

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

John H. Mathews and Russell W. Howell, Complex Analysis for Mathematics and Engineering, 6^th edition, Jones and Bartlett, 2012.

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 (1.5 weeks)
Origin, algebra, geometry and topology of complex numbers

• Chapter 2: Complex Functions (2 weeks)
 Functions and linear mappings, power functions, the reciprocal function; Limits and Continuity; Branches of functions
• Chapter 3: Analytic and Harmonic Functions (2 weeks)
 Differentiable and Analytic functions; Cauchy-Riemann equations; Harmonic functions.

• Chapter 4: Sequences, Julia and Manedlbrot Sets, and Power Series (1/2 week)
 Sequence and Series, Julia and Mandelbrot Sets (optional)
• Chapter 5: Elementary Functions (2 weeks)
 Complex Exponential function, Complex Logarithm, Complex Exponents, Trigonometric Functions
• Chapter 6: Complex Integration (2 weeks)
 Complex Integrals, Contours and Contour Integrals, Cauchy-Goursat Theorem, Fundamental Theorems of Integration, Integral Representations for Analytic Functions, Theorems of Morera and Liouville (if time)
• Chapter 7: Taylor and Laurent Series (1 week)
 Uniform Convergence; Taylor Series representations; Laurent Series representations; Singularities, Zeros, Poles
• Chapter 8: Residue Theory (1 week) Residue Theorem

Times above include review and testing.