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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 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.



Table of Contents:
Preface; Notation and abbreviations; Part I. Riemann-Hilbert Problems: 1. Classical applications of Riemann-Hilbert problems; 2. Riemann-Hilbert problems; 3. Inverse scattering and nonlinear steepest descent; Part II. Numerical Solution of Riemann-Hilbert Problems: 4. Approximating functions; 5. Numerical computation of Cauchy transforms; 6. The numerical solution of Riemann-Hilbert problems; 7. Uniform approximation theory for Riemann-Hilbert problems; Part III. The Computation of Nonlinear Special Functions and Solutions of Nonlinear PDEs: 8. The Korteweg-de Vries and modified Korteweg-de Vries equations; 9. The focusing and defocusing nonlinear Schrodinger equations; 10. The Painleve II transcendents; 11. The finite-genus solutions of the Korteweg-de Vries equation; 12. The dressing method and nonlinear superposition; Part IV. Appendices: A. Function spaces and functional analysis; B. Fourier and Chebyshev series; C. Complex analysis; D. Rational approximation; E. Additional KdV results; Bibliography; Index.