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Introduction to Complex Analysis

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Prerequisites: Qualifying math score; successful completion of Multivariable Calculus and Introduction to Abstract Math or the equivalent.

Course Format: Individually Paced
Course Length: Typically 6 months
Recommended School Credit: One full year of high school credit or one semester of college credit equal to or greater than an AP class
Course Code: CPX

Course Description


Introduction to Complex Analysis gives advanced students an introduction to the theory of functions of a complex variable, a fundamental area of mathematics. Topics include complex numbers and their properties, analytic functions and the Cauchy-Riemann equations, the logarithm and other elementary functions of a complex variable, integration of complex functions, the Cauchy integral theorem and its consequences, power series representation of analytic functions, the residue theorem and applications to definite integrals. Definitions and proofs will be stressed throughout the course. Online course materials supplement the required textbook. The course will also cover applications to physics and engineering.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructor via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include homework, chapter exams, and a cumulative midterm and final. Instructors use virtual classroom software allowing video, voice, text, screen sharing and whiteboard interaction.

See the "List of Topics" tab for the complete listing of topics covered.

A Mandelbrot set.

Materials Needed

A textbook purchase is required for this course:

Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics (3rd Edition), by Edward B. Saff and A.D. Snider. 

  • ISBN-13: 978-0139078743
  • ISBN-10: 0139078746

Mandelbrot set

List of Topics

Complex Numbers

  • The Algebra of Complex Numbers
  • Point Representation of Complex Numbers
  • Vectors and Polar Forms
  • The Complex Exponential
  • Powers and Roots
  • Planar Sets
  • The Riemann Sphere and Stereographic Projection

Analytic Functions

  • Functions of a Complex Variable
  • Limits and Continuity
  • Analyticity
  • The Cauchy-Riemann Equations
  • Harmonic Functions
  • Steady-State Temperature as a Harmonic Function
  • Iterated Maps: Julia and Mandelbrot Sets

Elementary Functions

  • Polynomials and Rational Functions
  • The Exponential, Trigonometric, and Hyperbolic Functions
  • The Logarithmic Function
  • Washers, Wedges, and Walls
  • Complex Powers and Inverse Trigonometric Functions
  • Application to Oscillating Systems

Complex Integration

  • Contours
  • Contour Integrals
  • Independence of Path
  • Cauchy’s Integral Theorem
  • Deformation of Contours Approach
  • Vector Analysis Approach
  • Cauchy’s Integral Formula and Its Consequences
  • Bounds for Analytic Functions
  • Applications to Harmonic Functions

Series Representations for Analytic Functions

  • Sequences and Series
  • Taylor Series
  • Power Series
  • Mathematical Theory of Convergence
  • Laurent Series
  • Zeros and Singularities
  • The Point at Infinity
  • Analytic Continuation

Residue Theory

  • The Residue Theorem
  • Trigonometric Integrals over [0, 2π]
  • Improper Integrals of Certain Functions over (-, )
  • Improper Integrals Involving Trigonometric Functions
  • Indented Contours
  • Integrals Involving Multiple-Valued Functions
  • The Argument Principle and Rouché’s Theorem

Conformal Mapping

  • Invariance of Laplace’s Equation
  • Geometric Considerations
  • Möbius Transformations
  • Möbius Transformations, Continued
  • The Schwarz-Christoffel Transformation
  • Applications in Electrostatics, Heat Flow, and Fluid Mechanics
  • Further Physical Applications of Conformal Mapping

The Transforms of Applied Mathematics

  • Fourier Series (The Finite Fourier Transform)
  • The Fourier Transform
  • The Laplace Transform
  • The ζ-Transform
  • Cauchy Integrals and the Hilbert Transform

Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

This course uses an online virtual classroom for discussions with the instructor. The classroom works on standard computers with the Adobe Connect Add-in or Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile appStudents who are unable to attend live sessions will need a computer with the Adobe Connect Add-in or Adobe Flash plugin installed to watch recorded meetings. The Adobe Connect Add-in, Adobe Flash plugin, and Adobe Connect Mobile app are available for free download. Students who do not have the Flash plug-in installed or enabled on their browsers will be prompted to download and install the Adobe Connect add-in when accessing the virtual classroom.