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 provides introductions to both computational complex analysis, as well as to the applied theory of Riemann–Hilbert problems from an analytical and numerical perspective. Following a full-discussion of applications to integrable systems, differential equations and special function theory, the authors include six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann–Hilbert method, each of mathematical or physical significance, or both. As the most comprehensive book to date on the applied and computational theory of Riemann–Hilbert problems, this book is ideal for graduate students and researchers interested in a computational or analytical introduction to the Riemann–Hilbert method.

Thomas Trogdon is an NSF Postdoctoral Fellow at the Courant Institute of Mathematical Sciences, New York University. He was awarded the 2014 SIAM Richard C. DiPrima Prize for his dissertation, which shares its title with this book. He has published in the fields of numerical analysis, approximation theory, optical physics, integrable systems, partial differential equations and random matrix theory.

Sheehan Olver is a Senior Lecturer in the School of Mathematics and Statistics at the University of Sydney. Dr Olver was awarded the 2012 Adams Prize for his work on the numerical solution of Riemann–Hilbert problems. He has published in the fields of numerical analysis, approximation theory, integrable systems, oscillatory integrals, spectral methods and random matrix theory.