MATH 3364 - Introduction to Complex Analysis - University of Houston
Skip to main content

MATH 3364 - Introduction to Complex Analysis

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

Prerequisites: MATH 3331.

Course Description: The complex number system, analytic functions, the Cauchy integral theorem, series representation, residue theory, and conformal mapping

Text: Fundamentals of Complex Analysis with Applications to Engineering and Science, 3rd Edition, by E.B. Saff and A.D. Snider, Prentice-Hall, 2003. ISBN: 9780139078743


Chapter 1: Complex Numbers
1.1 The Algebra of Complex Numbers
1.2 Point Representation of Complex Numbers
1.3 Vectors and Polar Forms
1.4 The Complex Exponential
1.5 Powers and Roots
1.6 Planar Sets
1.7 The Riemann Sphere and Stereographic Projection

Chapter 2: Analytic Functions
2.1 Functions of a Complex Variable
2.2 Limits and Continuity
2.3 Analyticity
2.4 The Cauchy-Riemann Equations
2.5 Harmonic Functions

Chapter 3: Elementary Functions
3.1 Polynomials and Rational Functions
3.2 The Exponential, Trigonometric and Hyperbolic Functions
3.3 The Logarithmic Function
3.4 Washers, Wedges, and Walls
3.5 Complex Powers and Inverse Trigonometric Functions

Chapter 4: Complex Integration
4.1 Contours
4.2 Contour Integrals
4.3 Independence of Path
4.4 Cauchy's Integral Theorem
4.5 Cauchy's Integral Formula and Its Consequences
4.6 Bounds for Analytic Functions

Chapter 5: Series Representations for Analytic Functions
5.1 Sequences and Series
5.2 Taylor Series
5.3 Power Series
5.4 Mathematical Theory of Convergence
5.5 Laurent Series
5.6 Zeros and Singularities
5.7 The Point at Infinity

Chapter 6: Residue Theory
6.1 The Residue Theorem
6.2 Trigonometric Integrals
6.3 Improper Integrals of Certain Functions
6.4 Improper Integrals Involving Trigonometric Functions
6.5 Indented Contours
6.6 Integrals Involving Multiple-Value Functions
6.7 The Argument Principle and Rouche's Theorem
At the instructor's discretion, other topics as time permits.


Grading: Please consult your instructor's syllabus regarding any and all grading guidelines.

Justin Dart Jr. Center Accommodations:

Academic Adjustments/Auxiliary Aids: The University of Houston System complies with Section 504 of the Rehabilitation Act of 1973 and the Americans with Disabilities Act of 1990, pertaining to the provision of reasonable academic adjustments/auxiliary aids for students who have a disability. In accordance with Section 504 and ADA guidelines, University of Houston strives to provide reasonable academic adjustments/auxiliary aids to students who request and require them. If you believe that you have a disability requiring an academic adjustments/auxiliary aid, please visit Justin Dart Jr. Student Accessibility Center website at for more information.


Counseling and Psychological Services (CAPS) can help students who are having difficulties managing stress, adjusting to college, or feeling sad and hopeless. You can reach (CAPS) by calling 713-743-5454 during and after business hours for routine appointments or if you or someone you know is in crisis. No appointment is necessary for the "Let's Talk" program, a drop-in consultation service at convenient locations and hours around campus.