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Full Description
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.
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.
