Methods of Mathematical Physics Partial Differential Equations

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ISBN: 9780471504399 | 0471504394
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Copyright: 1/8/1991

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Since the first volume of this work came out in Germany in 1937, this book, together with its first volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's final revision of 1961.

I. Introductory Remarks

(61)

General Information about the Variety of Solutions

(10)

Examples

(6)

Differential Equations for Given Families of Functions

(4)

Systems of Differential Equations

(6)

The Question of Equivalence of a System of Differential Equations and a Single Differential Equation

(2)

Elimination from a Linear System with Constant Coefficients

(1)

Determined, Overdetermined, Underdetermined Systems

(3)

Methods of Integration for Special Differential Equations

(4)

Separation of Variables

(2)

Construction of Further Solutions by Superposition. Fundamental Solution of the Heat Equation. Poisson's Integral

(2)

Geometric Interpretation of a First Order Partial Differential Equation in Two Independent Variables. The Complete Integral

(6)

Geometric Interpretation of a First Order Partial Differential Equation

(2)

The Complete Integral

(1)

Singular Integrals

(2)

Examples

(1)

Theory of Linear and Quasi-Linear Differential Equations of First Order

(4)

Linear Differential Equations

(3)

Quasi-Linear Differential Equations

(1)

The Legendre Transformation

(7)

The Legendre Transformation for Functions of Two Variables

(2)

The Legendre Transformation for Functions of n Variables

(1)

Application of the Legendre Transformation to Partial Differential Equations

(4)

The Existence Theorem of Cauchy and Kowalewsky

(23)

Introduction and Examples

(4)

Reduction to a System of Quasi-Linear Differential Equations

(4)

Determination of Derivatives Along the Initial Manifold

(1)

Existence Proof for Solutions of Analytic Differential Equations

(7)

Observation About Linear Differential Equations

(1)

Remark About Nonanlytic Differential Equations

(1)

Remarks on Critical Initial Data. Characteristics

(3)

Appendix 1 to Chapter I. Laplace's Differential Equations for the Support Function of a Minimal Surface

(2)

Appendix 2 to Chapter I. Systems of Differential Equations of First Order and Differential Equations of Higher Order

(1)

Plausibility Considerations

(1)

Conditions of Equivalence for Systems of Two First Order Partial Differential Equations and a Differential Equation of Second Order

(4)

II. General Theory of Partial Differential Equations of First Order

(92)

Geometric Theory of Quasi-Linear Differential Equations in Two Independent Variables

(7)

Characteristic Curves

(2)

Initial Value Problem

(2)

Examples

(3)

Quasi-Linear Differential Equations in n Independent Variables

(6)

General Differential Equations in Two Independent Variables

(9)

Characteristic Curves and Focal Curves. The Monge Cone

(4)

Solution of the Initial Value Problem

(3)

Characteristics as Branch Elements. Supplementary Remarks. Integral Conoid. Caustics

(2)

The Complete Integral

(2)

Focal Curves and the Monge Equation

(2)

Examples

(9)

The Differential Equation of Straight Light Rays, (grad u)2 = 1

(3)

The Equation F (uz, uy) = 0

(3)

Clairaut's Differential Equation

(1)

Differential Equation of Tubular Surfaces

(1)

Homogeneity Relation

(1)

General Differential Equation in n Independent Variables

(6)

Complete Integral and Hamilton-Jacobi Theory

103

(10)

Construction of Envelopes and Characteristic Curves

103

(3)

Canonical Forms of the Characteristic Differential Equations

106

(1)

Hamilton-Jacobi Theory

107

(2)

Example. The Two-Body Problem

109

(2)

Example. Geodesics on an Ellipsoid

111

(2)

Hamilton-Jacobi Theory and the Calculus of Variations

113

(14)

Euler's Differential Equations in Canonical Form

114

(1)

Geodetic Distance or Eiconal and Its Derivatives. Hamilton-Jacobi Partial Differential Equation

115

(4)

Homogeneous Integrands

119

(2)

Fields of Extremals. Hamilton-Jacobi Differential Equation

121

(3)

Cone of Rays. Huyghens' Construction

124

(1)

Hilbert's Invariant Integral for the Representation of the Eiconal

125

(2)

Theorem of Hamilton and Jacobi

127

(1)

Canonical Transformations and Applications

127

(5)

The Canonical Transformation

127

(2)

New Proof of the Hamilton-Jacobi Theorem

129

(1)

Variation of Constants (Canonical Perturbation Theory)

130

(2)

Appendix 1 to Chapter II

132

(1)

Further Discussion of Characteristic Manifolds

132

(7)

Remarks on Differentiation in n Dimensions

132

(3)

Initial Value Problem. Characteristic Manifolds

135

(4)

Systems of Quasi-Linear Differential Equations with the Same Principal Part. New Derivation of the Theory

139

(6)

Haar's Uniqueness Proof

145

(9)

Appendix 2 to Chapter II. Theory of Conservation Laws

147

(7)

III. Differential Equation of HIgher Order

154

(86)

Normal Forms for Linear and Quasi-Linear Differential Operators of Second Order in Two Independent Variables

154

(16)

Elliptic, Hyperbolic, and Parabolic Normal Forms. Mixed Types

155

(5)

Examples

160

(3)

Normal Forms for Quasi-Linear Second Order Differential Equations in Two Variables

163

(4)

Example. Minimal Surfaces

167

(2)

Systems of Two Differential Equations of First Order

169

(1)

Classification in General and Characteristics

170

(10)

Notations

170

(1)

Systems of First Order with Two Independent Variables. Characteristics

171

(2)

Systems of First Order with n Independent Variables

173

(2)

Differential Equations of Higher Order. Hyperbolicity

175

(1)

Supplementary Remarks

176

(1)

Examples. Maxwell's and Dirac's Equations

176

(4)

Linear Differential Equations with Constant Coefficients

180

(17)

Normal Form and Classification for Equations of Second Order

181

(3)

Fundamental Solutions for Equations of Second Order

184

(3)

Plane Waves

187

(1)

Plane Waves Continued. Progressing Waves. Dispersion

188

(4)

Examples. Telegraph Equation. Undistorted Waves in Cables

192

(1)

Cylindrical and Spherical Waves

193

(4)

Initial Value Problems. Radiation Problems for the Wave Equation

197

(13)

Initial Value Problems for Heat Conduction. Transformation of the Theta Function

197

(4)

Initial Value Problems for the Wave Equation

201

(1)

Duhamel's Principle. Nonhomogeneous Equations. Retarded Potentials

202

(3)

Duhamel's Principle for Systems of First Order

204

(1)

Initial Value Problem for the Wave Equation in Two-Dimensional Space. Method of Descent

205

(1)

The Radiation Problem

206

(2)

Propagation Phenomena and Huyghens' Principle

208

(2)

Solution of Initial Value Problems by Fourier Integrals

210

(11)

Cauchy's Method of the Fourier Integral

210

(2)

Example

212

(3)

Justification of Cauchy's Method

215

(6)

Typical Problems in Differential Equations of Mathematical Physics

221

(11)

Introductory Remarks

221

(5)

Basic Principles

226

(4)

Remarks about ``Improperly Posed'' Problems

230

(1)

General Remarks About Linear Problems

231

(1)

Appendix 1 to Chapter III

232

(1)

Sobolev's Lemma

232

(2)

Adjoint Operators

234

(6)

Matrix Operators

234

(1)

Adjoint Differential Operators

235

(5)

Appendix 2 to Chapter III. The Uniqueness Theorem of Holmgren

237

(3)

IV. Potential Theory and Elliptic Differential Equations

240

(167)

Basic Notions

240

(21)

Equations of Laplace and Poisson, and Related Equations

240

(5)

Potentials of Mass Distributions

245

(7)

Green's Formulas and Applications

252

(6)

Derivatives of Potentials of Mass Distributions

258

(3)

Poisson's Integral and Applications

261

(14)

The Boundary Value Problem and Green's Function

261

(3)

Green's Function for the Circle and Sphere. Poisson's Integral for the Sphere and Half-Space

264

(4)

Consequences of Poisson's Formula

268

(7)

The Mean Value Theorem and Applications

275

(15)

The Homogeneous and Nonhomogeneous Mean Value Equation

275

(2)

The Converse of the Mean Value Theorems

277

(7)

Poisson's Equation for Potentials of Spatial Distributions

284

(2)

Mean Value Theorems for Other Elliptic Differential Equations

286

(4)

The Boundary Value Problem

290

(22)

Preliminaries. Continuous Dependence on the Boundary Values and on the Domain

290

(3)

Solution of the Boundary Value Problem by the Schwarz Alternating Procedure

293

(6)

The Method of Integral Equations for Plane Regions with Sufficiently Smooth Boundaries

299

(4)

Remarks on Boundary Values

303

(3)

Capacity and Assumption of Boundary Values

305

(1)

Perron's Method of Subharmonic Functions

306

(6)

The Reduced Wave Equations. Scattering

312

(8)

Background

312

(3)

Sommerfeld's Radiation Condition

315

(3)

Scattering

318

(2)

Boundary Value Problems for More General Elliptic Differential Equations. Uniqueness of the Solutions

320

(11)

Linear Differential Equations

320

(2)

Nonlinear Equations

322

(2)

Rellich's Theorem on the Monge-Ampere Differential Equation

324

(2)

The Maximum Principle and Applications

326

(5)

A Priori Estimates of Schauder and Their Applications

331

(19)

Schauder's Estimates

332

(4)

Solution of the Boundary Value Problem

336

(5)

Strong Barrier Functions and Applications

341

(2)

Some Properties of Solutions of L[u] = f

343

(3)

Further Results on Elliptic Equations; Behavior at the Boundary

346

(4)

Solution of the Beltrami Equations

350

(7)

The Boundary Value Problem for a Special Quasi-Linear Equation. Fixed Point Method of Leray and Schauder

357

(5)

Solution of Elliptic Differential Euquations by Means of Integral Equations

362

(13)

Construction of Particular Solutions. Fundamental Solutions. Parametrix

363

(4)

Further Remarks

367

(1)

Appendix to Chapter IV. Nonlinear Equations

367

(1)

Perturbation Theory

368

(1)

The Equation Δ = f(x, u)

369

(6)

Supplement to Chapter IV. Function Theoretic Aspects of the Theory of Elliptic Partial Differential Equations

374

(1)

Definition of Pseudoanalytic Functions

375

(2)

An Integral Equation

377

(1)

Similarity Principle

378

(4)

Applications of the Similarity Principle

382

(2)

Formal Powers

384

(2)

Differentiation and Integration of Pseudoanalytic Functions

386

(3)

Example. Equations of Mixed Type

389

(2)

General Definition of Pseudoanalytic Functions

391

(1)

Quasiconformality and a General Representation Theorem

392

(3)

A Nonlinear Boundary Value Problem

395

(4)

An Extension of Riemann's Mapping Theorem

399

(1)

Two Theorems on Minimal Surfaces

400

(1)

Equations with Analytic Coefficients

400

(1)

Proof of Privaloff's Theorem

401

(2)

Proof of the Schauder Fixed Point Theorem

403

(4)

V. Hyperbolic Differential Equations in Two Independent Variables

407

(144)

Introduction

407

(1)

Characteristics for Differential Equations Mainly of Second Order

408

(16)

Basic Notions. Quasi-Linear Equations

408

(6)

Characteristics on Integral Surfaces

414

(2)

Characteristics as Curves of Discontinuity. Wave Fronts. Propagation of Discontinuities

416

(2)

General Differential Equations of Second Order

418

(3)

Differential Equations of Higher Order

421

(2)

Invariance of Characteristics under Point Transformations

423

(1)

Reduction to Quasi-Linear Systems of First Order

423

(1)

Characteristic Normal Forms for Hyperbolic Systems of First Order

424

(5)

Linear, Semilinear and Quasi-Linear Systems

424

(3)

The Case k = 2. Linearization by the Hodograph Transformation

427

(2)

Applications to Dynamics of Compressible Fluids

429

(9)

One-Dimensional Isentropic Flow

429

(3)

Spherically Symmetric Flow

432

(1)

Steady Irrotational Flow

432

(2)

Systems of Three Equations for Nonisentropic Flow

434

(2)

Linearized Equations

436

(2)

Uniqueness. Domain of Dependence

438

(11)

Domains of Dependence, Influence, and Determinacy

438

(2)

Uniqueness Proofs for Linear Differential Equations of Second Order

440

(5)

General Uniqueness Theorem for Linear Systems of First Order

445

(3)

Uniqueness for Quasi-Linear Systems

448

(1)

Energy Inequalities

449

(1)

Riemann's Representation of Solutions

449

(12)

The Initial Value Problem

450

(1)

Riemann's Function

450

(4)

Symmetry of Riemann's Function

454

(1)

Riemann's Function and Radiation from a Point. Generalization to Higher Order Problems

455

(2)

Examples

457

(4)

Solution of Hyperbolic Linear and Semilinear Initial Value Problems by Iteration

461

(15)

Construction of the Solution for a Second Order Equation

462

(2)

Notations and Results for Linear and Semilinear Systems of First Order

464

(2)

Construction of the Solution

466

(4)

Remarks. Dependence of Solutions on Parameters

470

(1)

Mixed Initial and Boundary Value Problems

471

(5)

Cauchy's Problem for Quasi-Linear Systems

476

(2)

Cauchy's Problem for Single Hyperbolic Differential Equations of Higher Order

478

(8)

Reduction to a Characteristic System of First Order

479

(1)

Characteristic Representation of L[u]

480

(2)

Solution of Cauchy's Problem

482

(1)

Other Variants for the Solution. A Theorem

483

(2)

P. Ungar

Remarks

485

(1)

Discontinuities of Solutions. Shocks

486

(4)

Generalized Solutions. Weak Solutions

486

(2)

Discontinuities for Quasi-Linear Systems Expressing Conservation Laws. Shocks

488

(2)

Appendix 1 to Chapter V. Application of Characteristics as Coordinates

490

(1)

Additional Remarks on General Nonlinear Equations of Second Order

490

(5)

The Quasi-Linear Differential Equation

491

(3)

The General Nonlinear Equation

494

(1)

The Exceptional Character of the Monge-Ampere Equation

495

(4)

Transition from the Hyperbolic to the Elliptic Case Through Complex Domains

499

(2)

The Analyticity of the Solutions in the Elliptic Case

501

(4)

Function-Theoretic Remark

501

(1)

Analyticity of the Solutions of Δ = f (x, y, u, p, q)

502

(3)

Remark on the General Differential Equation F (x, y, u, p, q, r, s, t) = 0

505

(1)

Use of Complex Quantities for the Continuation of Solutions

505

(3)

Appendix 2 to Chapter V. Transient Problems and Heaviside Operational Calculus

507

(1)

Solution of Transient Problems by Integral Representation

508

(9)

Explicit Example. The Wave Equation

508

(3)

General Formulation of the Problem

511

(1)

The Integral of Duhamel

512

(3)

Method of Superposition of Exponential Solutions

515

(2)

The Heaviside Method of Operators

517

(18)

The Simplest Operators

518

(3)

Examples of Operators and Applications

521

(4)

Applications to Heat Conduction

525

(2)

Wave Equation

527

(1)

Justification of the Operational Calculus. Interpretation of Further Operators

528

(7)

General Theory of Transient Problems

535

(16)

The Laplace Transformation

535

(4)

Solution of Transient Problems by the Laplace Transformation

539

(5)

Example. The Wave and Telegraph Equations

544

(7)

VI. Hyperbolic Differential Equations in More Than Two Independent Variables

551

(1)

Introduction

551

(1)

PART I. Uniqueness, Construction, and Geometry of Solutions

552

(124)

Differential Equations of Second Order. Geometry of Characteristics

552

(17)

Quasi-Linear Differential Equations of Second Order

552

(4)

Linear Differential Equations

556

(2)

Rays or Bicharacteristics

558

(2)

Characteristics as Wave Fronts

560

(1)

Invariance of Characteristics

561

(2)

Ray Cone, Normal Cone, and Ray Conoid

563

(1)

Connection with a Riemann Metric

564

(2)

Reciprocal Transformations

566

(2)

Huyghens' Construction of Wave Fronts

568

(1)

Space-Like Surfaces. Time-Like Directions

569

(1)

Second Order Equations. The Role of Characteristics

569

(8)

Discontinuities of Second Order

570

(1)

The Differential Equation along a Characteristic Surface

571

(2)

Propagation of discontinuities along Rays

573

(1)

Illustration. Solution of Cauchy's Problem for the Wave Equation in Three Space Dimensions

574

(3)

Geometry of Characteristics for Higher Order Operators

577

(20)

Notation

577

(2)

Characteristic Surfaces, Forms, and Matrices

579

(2)

Interpretation of the Characteristic Condition in Time and Space. Normal Cone and Normal Surface. Characteristic Nullvectors and Eigenvalues

581

(2)

Construction of Characteristic Surfaces or Fronts. Rays, Ray Cone, Ray Conoid

583

(2)

Wave Fronts and Huyghens' Construction. Ray Surface and Normal Surfaces

585

(4)

Example

588

(1)

Invariance Properties

589

(1)

Hyperbolicity. Space-Like Manifolds, Time-Like Directions

589

(4)

Symmetric Hyperbolic Operators

593

(1)

Symmetric Hyperbolic Equations of Higher Order

594

(1)

Multiple Characteristic Sheets and Reducibility

595

(2)

Lemma on Bicharacteristic Directors

597

(21)

Examples. Hydrodynamics, Crystal Optics, Magnetohydrodynamics

599

(1)

Introduction

599

(1)

The Differential Equation System of Hydrodynamics

600

(2)

Crystal Optics

602

(3)

The Shapes of the Normal and Ray Surfaces

605

(4)

Cauchy's Problem for Crystal Optics

609

(3)

Magnetohydrodynamics

612

(6)

Propagation of Discontinuities and Cauchy's Problem

618

(18)

Introduction

618

(1)

Discontinuities of First Derivatives for Systems of First Order. Transport Equation

618

(2)

Discontinuities of Initial Values. Introduction of Ideal Functions. Progressing Waves

620

(4)

Propagation of Discontinuities for Systems of First Order

624

(2)

Characteristics with Constant Multiplicity

626

(3)

Examples for Propagation of Discontinuities Along Manifolds of More Than One Dimension. Conical Refraction

627

(2)

Resolution of Initial Discontinuities and Solution of Cauchy's Problem

629

(3)

Characteristic Surfaces as Wave Fronts

631

(1)

Solution of Cauchy's Problem by Convergent Wave Expansions

632

(1)

Systems of Second and Higher Order

633

(2)

Supplementary Remarks. Weak Solutions. Shocks

635

(1)

Oscillatory Initial Values. Asymptotic Expansion of the Solution. Transition to Geometrical Optics

636

(6)

Preliminary Remarks. Progressing Waves of Higher Order

636

(1)

Construction of Asymptotic Solutions

637

(3)

Geometrical Optics

640

(2)

Examples of Uniqueness Theorems and Domain of Dependence for Initial Value Problems

642

(7)

The Wave Equation

643

(3)

The Differential Equation (Darboux Equation)

646

(1)

Maxwell's Equations in Vacuum

647

(2)

Domains of Dependence for Hyperbolic Problems

649

(3)

Introduction

649

(1)

Description of the Domain of Dependence

650

(2)

Energy Integrals and Uniqueness for Linear Symmetric Hyperbolic Systems of First Order

652

(9)

Energy Integrals and Uniqueness for the Cauchy Problem

652

(2)

Energy Integrals of First and Higher Order

654

(2)

Energy Inequalities for Mixed Initial and Boundary Value Problems

656

(3)

Energy Integrals for Single Second Order Equations

659

(2)

Energy Estimates for Equations of Higher Order

661

(7)

Introduction

661

(1)

Energy Identities and Inequalities for Solutions of Higher Order Hyperbolic Operators. Method of Leray and Garding

662

(4)

Other Methods

666

(2)

The Existence Theorem

668

(8)

Introduction

668

(1)

The Existence Theorem

669

(2)

Remarks on Persistence of Properties of Initial Values and on Corresponding Semigroups. Huyghens' Minor Principle

671

(2)

Focussing. Example of Nonpersistence of Differentiability

673

(2)

Remarks about Quasi-Linear Systems

675

(1)

Remarks about Problems of Higher Order or Nonsymmetric Systems

675

(1)

PART II. Representation of Solutions

676

(123)

Introduction

676

(5)

Outline. Notations

676

(1)

Some Integral Formulas. Decomposition of Functions into Plane Waves

677

(4)

Equations of Second Order with Constant Coefficients

681

(18)

Cauchy's Problem

681

(2)

Construction of the Solution for the Wave Equation

683

(3)

Method of Descent

686

(2)

Further Discussion of the Solution. Huyghens' Principle

688

(3)

The Nonhomogeneous Equation. Duhamel's Integral

691

(1)

Cauchy's Problem for the General Linear Equation of Second Order

692

(3)

The Radiation Problem

695

(4)

Method of Spherical Means. The Wave Equation and the Darboux Equation

699

(7)

Darboux's Differential Equation for Mean Values

699

(1)

Connection with the Wave Equation

700

(3)

The Radiation Problem of the Wave Equation

703

(1)

Generalized Progressing Spherical Waves

704

(2)

The Initial Value Problem for Elastic Waves Solved by Spherical Means

706

(5)

Method of Plane Mean Values. Application to General Hyperbolic Equations with Constant Coefficients

711

(7)

General Method

711

(4)

Application to the Solution of the Wave Equation

715

(3)

Application to the Equations of Crystal Optics and Other Equations of Fourth Order

718

(9)

Solution of Cauchy's Problem

718

(5)

Further Discussion of the Solution. Domain of Dependence. Gaps

723

(4)

The Solution of Cauchy's Problem as Linear Functional of the Data. Fundamental Solutions

727

(17)

Description. Notations

727

(3)

Construction of the Radiation Function by Decomposition of the Delta Function

730

(3)

Regularity of the Radiation Matrix

733

(3)

The Generalized Huyghens Principle

735

(1)

Example. Special Linear Systems with Constant Coefficients. Theorem on Gaps

736

(1)

Example. The Wave Equation

737

(3)

Example. Hadamard's Theory for Single Equations of Second Order

740

(4)

Further Examples. Two Independent Variables. Remarks

744

(1)

Ultrahyperbolic Differential Equations and General Differential Equations of Second Order with Constant Coefficients

744

(10)

The General Mean Value Theorem of Asgeirsson

744

(4)

Another Proof of the Mean Value Theorem

748

(1)

Application to the Wave Equation

749

(1)

Solutions of the Characteristic Initial Value Problem for the Wave Equation

749

(3)

Other Applications. The Mean Value Theorem for Confocal Ellipsoids

752

(2)

Initial Value Problems for Non-Space-Like Initial Manifolds

754

(6)

Functions Determined by Mean Values over Spheres with Centers in a Plane

754

(2)

Applications to the Initial Value Problem

756

(4)

Remarks About Progressing Waves, Transmission of Signals and Huyghens' Principle

760

(6)

Distortion-Free Progressing Waves

760

(3)

Spherical Waves

763

(1)

Radiation and Huyghens' Principle

764

(2)

Appendix to Chapter VI. Ideal Functions or Distributions

766

(1)

Underlying Definitions and Concepts

766

(8)

Introduction

766

(1)

Ideal Elements

767

(1)

Notations and Definitions

768

(1)

Iterated Integration

769

(1)

Linear Functionals and Operators. Bilinear Form

769

(2)

Continuity of Functionals. Support of Test Functions

771

(1)

Lemma About r-Continuity

772

(1)

Some Auxiliary Functions

773

(1)

Examples

774

(1)

Ideal Functions

774

(14)

Introduction

774

(1)

Definition by Linear Differential Operators

775

(2)

Definition by Weak Limits

777

(1)

Definition by Linear Functionals

778

(1)

Equivalence. Representation of Functionals

779

(2)

Some Conclusions

781

(1)

Example. The Delta-Function

781

(2)

Identification of Ideal with Ordinary Functions

783

(2)

Definite Integrals. Finite Parts

785

(3)

Calculus with Ideal Functions

788

(4)

Linear Processes

789

(1)

Change of Independent Variables

789

(1)

Examples. Transformations of the Delta-Function

790

(1)

Multiplication and Convolution of Ideal Functions

791

(1)

Additional Remarks. Modifications of the Theory

792

(7)

Introduction

792

(1)

Different Spaces of Test Functions. The Space. Fourier Transforms

793

(2)

Periodic Functions

795

(1)

Ideal Functions and Hilbert Spaces. Negative Norms. Strong Definitions

796

(1)

Remark on Other Classes of Ideal Functions

797

(2)

Bibliography

799

(20)

Index

819

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Methods of Mathematical Physics Partial Differential Equations

Summary

Author Biography

Table of Contents

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