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

### Thomas Trogdon; Sheehan Olver

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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, and a discussion of applications to integrable systems, differential equations, and special function theory. It also includes six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann-Hilbert method, each of mathematical or physical significance or both.

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

Contents: 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 Schrödinger Equations; Chapter 10: The Painlevé 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

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Riemann-Hilbert problems are fundamental objects of study within complex analysis. 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 currently an NSF Postdoctoral Fellow at the Courant Institute of Mathematical Sciences at 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 currently 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.

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Book Description Society for Industrial Applied Mathematics,U.S., United States, 2016. Paperback. Book Condition: New. Language: English . Brand New Book. 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. Bookseller Inventory # AAN9781611974195

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Book Description Society for Industrial Applied Mathematics,U.S., United States, 2016. Paperback. Book Condition: New. Language: English . Brand New Book. 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. Bookseller Inventory # AAN9781611974195

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