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Introduction to Complex Analysis

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Eligibility: CTY-level or Advanced CTY-level math score required

Prerequisites: 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.

This course does not have any synchronous class meetings, but students may schedule one-on-one virtual meetings directly with the instructor to answer questions or concerns. Videos from YouTube or other web providers may be present in the course. Video recommendations or links provided at end of videos are generated by the video host provider and are not CTY recommendations.

A Mandelbrot set.

Proctor Requirements

For enrollment starting on or after January 1, 2019, a proctor is required for this course if the student’s goal is to obtain a grade. Please review Proctor Requirements for more details.

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.

Zoom online virtual classroom
This course uses an online virtual classroom which can be used for instructor-student communication if the student has any questions about the course or curriculum. The classroom works on standard computers with the Zoom desktop client and also tablets or handhelds that support the Zoom Mobile app. Students will need a computer with the Zoom desktop client installed to watch any recorded meetings. The Zoom desktop client and Zoom Mobile App are both available for free download.