Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions


Riemann-Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann-Hilbert problem.

This book, the most comprehensive one to date on the applied and computational theory of Riemann-Hilbert problems, includes:

  • an introduction to computational complex analysis,
  • an introduction to the applied theory of Riemann-Hilbert problems from an analytical and numerical perspective,
  • a discussion of applications to integrable systems, differential equations, and special function theory, and six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann-Hilbert method, each of mathematical or physical significance or both.

  • You will find Mathematca code for these examples on this webpage along with a collection of errata. See the RHPackage section below for preliminaries. Then you will find Mathematica notebooks to solve the Riemann-Hilbert problems that are discussed in Chapter 1, Sections 5.3, 5.5, 6.3, 6.4, and Chapters 8-11.

    This book is intended for graduate students and researchers interested in a computational or analytical introduction to the Riemann-Hilbert method.


The code presented on this page to solve Riemann-Hilbert problems requires RHPackage:
RHPackage Project Page
Once the RHPackage files are downloaded, we assume that all Mathematica notebooks are saved into the "RHPackage" directory which contains the "RiemannHilbert" directory. This will allow the Mathematica notebook to import RHPackage.

About the authors

Thomas Trogdon is currently an NSF Postdoctoral Fellow at the Courant Institute.

Sheehan Olver is currently a senior lecturer in the School of Mathematics at The University of Sydney

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