This is the main repository of course materials for MATH 6640 at RPI, Spring 2025. The syllabus is posted in the README below. Lecture notes, homework, exams, and supplementary materials will be posted here or linked through Piazza and Gradescope (RCS access only).
Course description (from RPI Catalog)
Review of basic complex variables theory; power series, analytic functions, singularities, and integration in the complex plane. Integral transforms (Laplace, Fourier, etc.) in the complex plane, with application to solution of PDEs and integral equations. Asymptotic expansions of integrals (Laplace method, methods of steepest descent and stationary phase), with emphasis on extraction of useful information from inversion integrals of transforms. Problems to be drawn from linear models in science and engineering.
Prerequisites: differential equations (MATH 2400), complex variables (MATH 4300), and advanced calculus (MATH 4600).
Lectures: Monday/Thursday 10–11:50am in Carnegie 106.
Instructor: Andrew Horning
Office Hours: Monday/Thursday 12:30-1:30pm in Amos Eaton 329.
Contact: hornia3@rpi.edu
Textbook: There is no required textbook, however, you may find the "further reading" suggested after each lecture summary below helpful in your studies. These will be drawn from Complex Analysis by Stein & Shakarchi (good high-level theoretical development), Complex Variables: Introduction and Applications (2nd Edition) by Ablowitz and Fokas (focused on applications, especially asymptotics), Applied Complex Variables by Dettman (detailed rigorous treatment of theory and applications). I also highly recommend Elias Wegart's phase plotting software in MATLAB and his accompanying book as tools for visualizing functions of complex variables.
Course Tools: Communicate (announcements, questions, and discussion) through Piazza. View and submit homework assignments on Gradescope. The mid-term will be in-class and the final project will be submitted on Gradescope.
Grading: 40% homework (due weekly on Friday at 5pm), 30% mid-term (February 27), 30% final project (April 23). Problem sets must be submitted to Gradescope before the deadline on the due date. Regrade requests can be made on Gradescope within one week after grades are published.
Collaboration and Academic Integrity: To maximize your learning objectives, reserve time to work on each problem independently before discussing it with your classmates. Always write up the solution on your own and acknowledge your collaborators. Copying solutions directly from peers, books, internet sources, or AI tools is strictly prohibited.
Accomodations and Disability Services: If you have approved accommodations through the Office of Disability Services for Students (DSS), please reach out to meet with me early in the semester. We are committed to equal access for all students and will be happy to facilitate the use of approved accommodations.
Instead of a final exam, write an 8-10 page report that reviews a topic in complex analysis or integral transforms. The scope of the report must extend beyond material covered in lecture or on problem sets. Your report should include three basic elements:
Review: Why is this topic interesting, what is its history, and what are the important publications and references? (A reasonably comprehensive bibliography is expected: not just the sources you happened to consult, but a curated collection of materials that you could recommend to a peer who wanted to learn more.)
Technical Survey: A concise description of the key mathematical question(s), technique(s), and/or result(s) connected to your topic. This should be written in clear and precise mathematical language, at a level accessible to your peers in this class.
Examples: Illustrate the basic principles (problems, techniques, results) with concrete examples. These could be introduced within the technical survey portion to illustrate key concepts or placed in a separate section (or both). Graphical illustrations are highly encouraged when possible. Computational experiments are also encouraged if appropriate for your topic (but not required).
Below are a few broad areas to explore for your project, with modern references on specific topics to help you get started.
- Contour Integral methods in Numerical Linear Algebra
- FEAST Eigensolver.
- Matrix Functions.
- PDE Solvers. Check out this Chebfun example and references at the end.
- Riemann--Hilbert Problems
- Analytic Continuation
- Conformal Mapping
- Fluid Flows. This is a beautiful collection of explainer videos and research articles from Peter Baddoo, whose recent thesis on this topic was awarded the "Best Thesis Award" by the UK Fluids Network and published as a book by Springer. You may also want to check out his former PhD advisor's work, Lorna Ayton at University of Cambridge.)
- Computing Conformal Maps. Check out the references at the end of this Chebfun example along with Toby Driscoll's Schwarz-Christoffel toolbox.
- Potential Theory
- Homework 1 is due by 5pm on
Friday, January 17Friday, January 24. (Solutions) - Homework 2 is due by 5pm on
Friday, January 31Tuesday, February 4. (Solutions) - Homework 3 is due by 5pm on Friday, February 14. (Solutions)
- Homework_4 is due by 5pm on Friday, February 21. (Solutions)
- Homework 5 is due by 5pm on Friday, March 21. (Solutions)
- Homework 6 is due by 5pm on Friday, April 11. (Solutions)
- Why do we need complex numbers?
- Three thematic examples
- The algebra of complex numbers
Notes | Chapter 1, Section 1 (Stein and Shakarchi)
- Functions of a complex variable
- Differentiating complex functions
- Cauchy-Riemann equations and power series
Notes | Chapter 1, Section 2 (Stein and Shakarchi) | Visualizing Complex Functions (article and software)
- Convergence and differentiability of power series
- Contour integration in the complex plane
- Cauchy's theorem and first applications
Notes | Chapter 1, Section 3 and Chapter 2, Sections 1-3 (Stein and Shakarchi)
- Cauchy's theorem vs Green's theorem
- Contour deformation strategies
- Cauchy's integral formula
Notes | Chapter 2.4-2.5 (Ablowitz and Fokas)
- Cauchy's integral formulas and their implications
- Cauchy's inequalities and power series representations
- Zeros of holomorphic functions and analytic continuation
Notes | Chapter 2, Sections 4-5 (Stein and Shakarchi)
- Summary: why are holomorphic functions like Tolstoy's "happy families"?
- Contour deformation "through" an isolated singularity
- Laurent series: expansion around an isolated singularity
Notes | Chapter 3.3 (Ablowitz and Fokas) and Chapter 3, Section 1 (Stein and Shakarchi).
- The "nature" of isolated singularities
- The residue of an isolated singularity
- The residue theorem
Notes | Chapters 3.5 (up to p. 150), 4.1 (Ablowitz and Fokas) and Chapter 3, Sections 1-3 (Stein and Shakarchi)
- What makes 1/z special?
- Complex antiderivatives
- The complex logarithm
Notes | Chapter 3, Section 6 (Stein and Shakarchi)
- Logarithms, arguments, and winding numbers
- The argument principle
- Rouche's theorem
Notes | Chapter 3, Section 4 (Stein and Shakarchi)
- Periodic signals and Fourier series
- Fourier coefficients from equispaced samples
- The exponentially convergent trapezoid rule
Notes | The exponentially convergent trapezoidal rule | Zolotarev's Problems
- Trapezoid rules and Discrete Fourier Transforms
- Reconstructing signals from Fourier coefficients
- Fourier coefficients of smooth periodic signals
- Fourier series and boundary value problems
- Differentiation and multiplication of Fourier series
- Fourier pseudospectral methods
- Heated ring: Fourier series solution
- Heated wire: Fourier transform solution
- The Fourier transform and its inverse
- Fourier Inversion
- Example: Heat Kernel
- The roles of decay and regularity
Notes | Chapter 4, Section 1 (Stein and Shakarchi)
- Fourier inversion theorems
- A complex analytic proof
- Bandlimited functions
Notes | Chapter 4, Section 2 (Stein and Shakarchi)
- Analytic continuation via Fourier transform
- Which functions are bandlimited? Which functions are "nearly" bandlimited?
- Paley-Wiener Theorem
Notes | Chapter 4, Section 3 (Stein and Shakarchi)
- Proof of Paley-Wiener Theorem
- Where and when do entire functions "blow up"?
- Phragmen-Lindelof Theorem
Notes | Chapter 4, Section 3 (Stein and Shakarchi)
- Hardy's Uncertainty Principle
- Application to quantum particle
- Three challenge problems
Notes | Chapter 4, Exercise 12 and Problem 3
- Extrapolation and model complexity
- Extrapolation of analytic functions
- Stable extrapolation of analytic functions
Notes | See Trefethen's "Quantifying the ill-conditioning of analytic continuation" for a clear and concise treatment of analytic continuation from inexact data using Hadamard's "Three Lines Lemma." See Demanet and Townsend's "Stable extrapolation of analytic functions" for an analysis of extrapolation from noisy samples using Chebyshev polynomials.
- The harmonic oscillator in the Fourier domain
- Constructing solutions in the complex Fourier domain
- Solution operators and Green's functions
- Green's functions for the harmonic oscillator
- The homogeneous equation and families of Green's functions
- Causal and anti-causal Green's functions
Notes | See Chapter 8.5 in Dettman for a more detailed description of Fourier transform solutions to constant-coefficient ODEs. See Chapter 7.3 in Introduction to Partial Differential Equations by Olver for another concise illustration of these techniques.
- The physics of anti-causal solutions
- Green's functions in the Fourier domain
- Poles, Contours, and Causality
- Homogeneous solutions and auxillary conditions
- Homogeneous solutions and residues
- Contour selection for auxillary conditions
- Linear constant coefficient differential operators
- The Laplace transform and initial value problems
- The Bromwich contour and the inverse Laplace transform
Notes | Chapter 9, Sections 1 and 2 (Dettman)
- Fourier analysis vs. Laplace analysis
- Structure of solutions in Laplace domain
- Linear Time Invariant (LTI) systems
Notes | Chapter 9, Sections 3 and 4 (Dettman)