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This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field.Originally published in 1972.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Mathematical potential theory --- Holomorphic functions --- Harmonic functions --- Holomorphic functions. --- Harmonic functions. --- Fonctions de plusieurs variables complexes. --- Functions of several complex variables --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Functions, Holomorphic --- Absolute continuity. --- Absolute value. --- Addition. --- Ambient space. --- Analytic function. --- Arbitrarily large. --- Bergman metric. --- Borel measure. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Boundedness. --- Brownian motion. --- Calculation. --- Change of variables. --- Characteristic function (probability theory). --- Combination. --- Compact space. --- Complex analysis. --- Complex conjugate. --- Computation. --- Conformal map. --- Constant term. --- Continuous function. --- Coordinate system. --- Corollary. --- Cramer's rule. --- Determinant. --- Diameter. --- Dimension. --- Elliptic operator. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Fatou's theorem. --- Function space. --- Green's function. --- Green's theorem. --- Haar measure. --- Half-space (geometry). --- Harmonic function. --- Hilbert space. --- Holomorphic function. --- Hyperbolic space. --- Hypersurface. --- Hölder's inequality. --- Invariant measure. --- Invertible matrix. --- Jacobian matrix and determinant. --- Line segment. --- Linear map. --- Lipschitz continuity. --- Local coordinates. --- Logarithm. --- Majorization. --- Matrix (mathematics). --- Maximal function. --- Measure (mathematics). --- Minimum distance. --- Natural number. --- Normal (geometry). --- Open set. --- Order of magnitude. --- Orthogonal complement. --- Orthonormal basis. --- Parameter. --- Poisson kernel. --- Positive-definite matrix. --- Potential theory. --- Projection (linear algebra). --- Quadratic form. --- Quantity. --- Real structure. --- Requirement. --- Scientific notation. --- Sesquilinear form. --- Several complex variables. --- Sign (mathematics). --- Smoothness. --- Subgroup. --- Subharmonic function. --- Subsequence. --- Subset. --- Summation. --- Tangent space. --- Theorem. --- Theory. --- Total variation. --- Transitive relation. --- Transitivity. --- Transpose. --- Two-form. --- Unit sphere. --- Unitary matrix. --- Vector field. --- Vector space. --- Volume element. --- Weak topology.

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This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

Schrodinger equation. --- Wave mechanics. --- Equation, Schrödinger --- Schrödinger wave equation --- Electrodynamics --- Matrix mechanics --- Mechanics --- Molecular dynamics --- Quantum statistics --- Quantum theory --- Waves --- Differential equations, Partial --- Particles (Nuclear physics) --- Wave mechanics --- WKB approximation --- Schrödinger equation. --- Schrödinger, Équation de. --- Solitons. --- Abelian integral. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation. --- Asymptote. --- Asymptotic analysis. --- Asymptotic distribution. --- Asymptotic expansion. --- Banach algebra. --- Basis (linear algebra). --- Boundary (topology). --- Boundary value problem. --- Bounded operator. --- Calculation. --- Cauchy's integral formula. --- Cauchy's integral theorem. --- Cauchy's theorem (geometry). --- Cauchy–Riemann equations. --- Change of variables. --- Coefficient. --- Complex plane. --- Cramer's rule. --- Degeneracy (mathematics). --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differential equation. --- Differential operator. --- Dirac equation. --- Disjoint union. --- Divisor. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Energy minimization. --- Equation. --- Euler's formula. --- Euler–Lagrange equation. --- Existential quantification. --- Explicit formulae (L-function). --- Fourier transform. --- Fredholm theory. --- Function (mathematics). --- Gauge theory. --- Heteroclinic orbit. --- Hilbert transform. --- Identity matrix. --- Implicit function theorem. --- Implicit function. --- Infimum and supremum. --- Initial value problem. --- Integrable system. --- Integral curve. --- Integral equation. --- Inverse problem. --- Jacobian matrix and determinant. --- Kerr effect. --- Laurent series. --- Limit point. --- Line (geometry). --- Linear equation. --- Linear space (geometry). --- Logarithmic derivative. --- Lp space. --- Minor (linear algebra). --- Monotonic function. --- Neumann series. --- Normalization property (abstract rewriting). --- Numerical integration. --- Ordinary differential equation. --- Orthogonal polynomials. --- Parameter. --- Parametrix. --- Paraxial approximation. --- Parity (mathematics). --- Partial derivative. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Perturbation theory. --- Pole (complex analysis). --- Polynomial. --- Probability measure. --- Quadratic differential. --- Quadratic programming. --- Radon–Nikodym theorem. --- Reflection coefficient. --- Riemann surface. --- Simultaneous equations. --- Sobolev space. --- Soliton. --- Special case. --- Taylor series. --- Theorem. --- Theory. --- Trace (linear algebra). --- Upper half-plane. --- Variational method (quantum mechanics). --- Variational principle. --- WKB approximation. --- Schrödinger, Équation de.