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. Six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann-Hilbert method, each of mathematical or physical significance or both.
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.
Chapter 1: Classical Applications of Riemann-Hilbert Problems Chapter 2: Riemann-Hilbert Problems Chapter 3: Inverse Scattering and Nonlinear Steepest Descent Chapter 4: Approximating Functions Chapter 5: Numerical Computation of Cauchy Transforms Chapter 6: The Numerical Solution of Riemann-Hilbert Problems Chapter 7: Uniform Approximation Theory for Riemann-Hilbert Problems Chapter 8: The Korteweg-de Vries and Modified Korteweg-de Vries Equations Chapter 9: The Focusing and Defocusing Nonlinear Schroedinger Equations Chapter 10: The Painleve II Transcendents Chapter 11: The Finite-Genus Solutions of the Korteweg-de Vries Equation Chapter 12: The Dressing Method and Nonlinear Superposition Appendix A: Function Spaces and Functional Analysis Appendix B: Fourier and Chebyshev Series Appendix C: Complex Analysis Appendix D: Rational Approximation Appendix E: Additional KdV Results.
The most comprehensive book to date on the applied and computational theory of Riemann-Hilbert problems, ideal for graduates and researchers.
