Methods of Mathematical Physics, Volume 1

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ISBN: 9780471504474 | 0471504475
Cover: Paperback
Copyright: 1/8/1991

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Since the first volume of this work came out in Germany in 1924, this book, together with its second 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 second and final revision of 1953.

I. The Algebra of Linear Transformations and Quadratic Forms

(47)

Linear equations and linear transformations

(16)

Vectors

(2)

Orthogonal systems of vectors. Completeness

(2)

Linear transformations. Matrices

(6)

Bilinear, quadratic, and Hermitian forms

(3)

Orthogonal and unitary transformations

(3)

Linear transformations with a linear parameter

(6)

Transformation to principal axes of quadratic and Hermitian forms

(8)

Transformation to principal axes on the basis of a maximum principle

(3)

Eigenvalues

(2)

Generalization to Hermitian forms

(1)

Intertial theorem for quadratic forms

(1)

Representation of the resolvent of a form

(1)

Solution of systems of linear equations associated with forms

(1)

Minimum-maximum property of eigenvalues

(3)

Characterization of eigenvalues by a minimum-maximum problem

(2)

Applications. Constraints

(1)

Supplement and problems

(14)

Linear Independence and the Gram determinant

(2)

Hadamard's inequality for determinants

(1)

Generalized treatment of canonical transformations

(4)

Bilinear and quadratic forms of infinitely many variables

(1)

Infinitesimal linear transformations

(1)

Perturbations

(2)

Constraints

(1)

Elementary divisors of a matrix or a bilinear form

(1)

Spectrum of a unitary matrix

(2)

References

(1)

II. Series Expansions of Arbitrary Functions

(64)

Orthogonal systems of functions

(8)

Definitions

(1)

Orthogonalization of functions

(1)

Bessel's inequality. Completeness relation. Approximation in the mean

(4)

Orthogonal and unitary transformations with infinitely many variables

(1)

Validity of the results for several independent variables. More general assumptions

(1)

Construction of complete systems of functions of several variables

(1)

The accumulation principle for functions

(4)

Convergence in function space

(4)

Measure of independence and dimension number

(4)

Measure of independence

(2)

Asymptotic dimension of a sequence of functions

(2)

Weierstrass's approximation theorem. Completeness of powers and of trigonometric functions

(4)

Weierstrass's approximation theorem

(3)

Extension to functions of several variables

(1)

Simultaneous approximation of derivatives

(1)

Completeness of the trigonometric functions

(1)

Fourier series

(8)

Proof of the fundamental theorem

(4)

Multiple Fourier series

(1)

Order of magnitude of Fourier coefficients

(1)

Change in length of basic interval

(1)

Examples

(3)

The Fourier integral

(4)

The fundamental theorem

(2)

Extension of the result to several variables

(1)

Reciprocity formulas

(1)

Examples of Fourier integrals

(1)

Legendre Polynomials

(5)

Construction of the Legendre polynomials by orthogonaliza-of the powers 1, x, x2

(3)

The generating function

(1)

Other properties of the Legendre polynomials

(1)

Examples of other orthogonal systems

(10)

Generalization of the problem leading to Legendre polynomials

(1)

Tehebycheff polynomials

(2)

Jacobi polynomials

(1)

Hermite polynomials

(2)

Laguerre polynomials

(2)

Completeness of the Laguerre and Hermite functions

(2)

Supplement and problems

(15)

Hurwitz's solution of the isoperimetric problem

(1)

Reciprocity formulas

(1)

The Fourier integral and convergence in the mean

(1)

Spectral decomposition by Fourier series and integrals

(1)

Dense systems of functions

100

(2)

A Theorem of H. Muntz on the completeness of powers

102

(1)

Fejer's summation theorem

102

(1)

The Mellin inversion formulas

103

(2)

The Gibbs Phenomenon

105

(2)

A Theorem on Gram's determinant

107

(1)

Application of the Lebesgue integral

108

(4)

References

111

(1)

III. Linear Integral Equations

112

(52)

Introduction

112

(3)

Notation and basic concepts

112

(1)

Functions in integral representation

113

(1)

Degenerate kernels

114

(1)

Fredholm's theorems for degenerate kernels

115

(3)

Fredholm's theorems for arbitrary kernels

118

(4)

Symmetric kernels and their eigenvalues

122

(12)

Existence of an eigenvalue of a symmetric kernel

122

(4)

The totality of eigenfunctions and eigenvalues

126

(6)

Maximum-minimum property of eigenvalues

132

(2)

The expansion theorem and its applications

134

(6)

Expansion theorem

134

(2)

Solution of the inhomogeneous linear integral equation

136

(1)

Bilinear formula for iterated kernels

137

(1)

Mercer's theorem

138

(2)

Neumann series and the reciprocal kernel

140

(2)

The Fredholm formulas

142

(5)

Another derivation of the theory

147

(5)

A Lemma

147

(1)

Eigenfunctions of a symmetric kernel

148

(2)

Unsymmetric Kernels

150

(1)

Continuous dependence of eigenvalues and eigenfunctions on the kernel

151

(1)

Extensions of the theory

152

(1)

Supplement and problems for Chapter III

153

(11)

Problems

153

(1)

Singular integral equations

154

(1)

E. Schmidt's derivation of the Fredholm theorems

155

(1)

Enskog's method for solving symmetric integral equations

156

(1)

Kellogg's method for the determination of eigenfunctions

156

(1)

Symbolic functions of a kernel and their eigenvalues

157

(1)

Examples of an unsymmetric kernel without null solutions

157

(1)

Volterra integral equation

158

(1)

Abel's integral equation

158

(1)

Adjoint orthogonal systems belonging to an unsymmetric kernel

159

(1)

Integral equations of the first kind

159

(1)

Method of infinitely many variables

160

(1)

Minimum properties of eigenfunctions

161

(1)

Polar integral equations

161

(1)

Symmetrizable kernels

161

(1)

Determination of the resolvent kernel by functional equations

162

(1)

Continuity of definite kernels

162

(1)

Hammerstein's theorem

162

(2)

References

162

(2)

IV. The Calculus of Variations

164

(111)

Problems of the calculus of variations

164

(10)

Maxima and minima of functions

164

(3)

Functionals

167

(2)

Typical problems of the calculus of variations

169

(4)

Characteristic difficulties of the calculus of variations

173

(1)

Direct solutions

174

(9)

The Isoperimetric problem

174

(1)

The Rayleigh-Ritz method. Minimizing Sequences

175

(1)

Other Direct methods. Method of finite differences. Infinitely many variables

176

(6)

General remarks on direct methods of the calculus of variations

182

(1)

The Euler equations

183

(23)

``Simplest problem'' of the variational calculus

184

(3)

Several unknown functions

187

(3)

Higher derivatives

190

(1)

Several independent variables

191

(2)

Identical vanishing of the Euler differential expression

193

(3)

Euler equations in homogeneous form

196

(3)

Relaxing of conditions. Theorems of du Bois-Reymond and Haar

199

(6)

Variational problems and functional equations

205

(1)

Integration of the Euler differential equation

206

(2)

Boundary conditions

208

(6)

Natural boundary conditions for free boundaries

208

(3)

Geometrical problems. Transversality

211

(3)

The second variation and the Legendre condition

214

(2)

Variational problems with subsidiary conditions

216

(6)

Isoperimetric problems

216

(3)

Finite subsidiary conditions

219

(2)

Differential equations as subsidiary conditions

221

(1)

Invariant character of the Euler equations

222

(9)

The Euler expression as a gradient in function space. Invariance of the Euler expression

222

(2)

Transformation of Δu. Spherical coordinates

224

(2)

Ellipsoidal coordinates

226

(5)

Transformation of variational problems to canonical and involutory form

231

(11)

Transformation of an ordinary minimum problem with subsidiary conditions

231

(2)

Involutory transformation of the simplest variational problems

233

(5)

Transformation of variational problems to canonical form

238

(2)

Generalizations

240

(2)

Variational calculus and the differential equations of mathematical physics

242

(10)

General remarks

242

(2)

The vibrating string and the vibrating rod

244

(2)

Membrane and plate

246

(6)

Reciprocal quadratic variational problems

252

(5)

Supplementary remarks and exercises

257

(18)

Variational problem for a given differential equation

257

(1)

Reciprocity for isoperimetric problems

258

(1)

Circular light rays

258

(1)

The problem of Dido

258

(1)

Examples of problems in space

258

(1)

The indicatrix and applications

258

(2)

Variable domains

260

(2)

E. Noether's theorem on invariant variational problems. Integrals in particle mechanics

262

(4)

Transversality for multiple integrals

266

(1)

Euler's differential expressions on surfaces

267

(1)

Thomson's principle in electrostatics

267

(1)

Equilibrium problems for elastic bodies. Castigliano's principle

268

(4)

The Variational problem of buckling

272

(3)

References

274

(1)

V. Vibration and Eigenvalue Problems

275

(122)

Preliminary remarks about linear differential equations

275

(6)

Principle of superposition

275

(2)

Homogeneous and nonhomogeneous problems. Boundary conditions

277

(1)

Formal relations. Adjoint differential expressions. Green's formulas

277

(3)

Linear functional equations as limiting cases and analogues of systems of linear equations

280

(1)

Systems of a finite number of degrees of freedom

281

(5)

Normal modes of vibration. Normal coordinates. General theory of motion

282

(3)

General properties of vibrating systems

285

(1)

The vibrating string

286

(9)

Free motion of the homogeneous string

287

(2)

Forced motion

289

(2)

The general nonhomogeneous string and the Sturm-Liouville eigenvalue problem

291

(4)

The vibrating rod

295

(2)

The vibrating membrane

297

(10)

General eigenvalue problem for the homogeneous membrane

297

(3)

Forced motion

300

(1)

Nodal lines

300

(1)

Rectangular membrane

300

(2)

Circular membrane. Bessel functions

302

(4)

Nonhomogeneous membrane

306

(1)

The Vibrating plate

307

(1)

General remarks

307

(1)

Circular boundary

307

(1)

General remarks on the eigenfunction method

308

(5)

Vibration and equilibrium problems

308

(3)

Heat conduction and eigenvalue problems

311

(2)

Vibration of three-dimensional continua. Separation of variables

313

(2)

Eigenfunctions and the boundary value problem of potential theory

315

(9)

Circle, sphere, and spherical shell

315

(4)

Cylindrical domain

319

(1)

The Lame problem

319

(5)

Problems of the Sturm-Liouville type. Singular boundary points

324

(7)

Bessel functions

324

(1)

Legendre functions of arbitrary order

325

(2)

Jacobi and tehebycheff polynomials

327

(1)

Hermite and Laguerre polynomials

328

(3)

The asymptotic behavior of the solutions of Strum-Liouville equations

331

(8)

Boundedness of the solution as the independent variables tends to infinity

331

(1)

A sharper result. (Bessel functions)

332

(2)

Boundedness as the parameter increases

334

(1)

Asymptotic representation of the solutions

335

(1)

Asymptotic representation of Sturm-Liouville eigenfunctions

336

(3)

Eigenvalue problems with a continuous spectrum

339

(4)

Trigonometric functions

340

(1)

Bessel functions

340

(1)

Eigenvalue problem of the membrane equation for the infinite plane

341

(1)

The Schrodinger eigenvalue problem

341

(2)

Perturbation theory

343

(8)

Simple eigenvalues

344

(2)

Multiple eigenvalues

346

(2)

An example

348

(3)

Green's function (influence function) and reduction of differential equations to integral equations

351

(20)

Green's function and boundary value problem for ordinary differential equations

351

(3)

Construction of Green's function; Green's function in the generalized sense

354

(4)

Equivalence of integral and differential equations

358

(4)

Ordinary differential equations of higher order

362

(1)

Partial differential equations

363

(8)

Examples of Green's function

371

(17)

Ordinary differential equations

371

(6)

Green's function for Δu: circle and sphere

377

(1)

Green's function and conformal mapping

377

(1)

Green's function for the potential equation on the surface of a sphere

378

(1)

Green's function for Δ u = 0 in a rectangular parallelepiped

378

(6)

Green's function for Δ u in the interior of a rectangle

384

(2)

Green's function for a circular ring

386

(2)

Supplement to Chapter V

388

(9)

Examples for the vibrating string

388

(2)

Vibrations of a freely suspended rope; Bessel functions

390

(1)

Examples for the explicit solution of the vibration equation. Mathieu functions

391

(1)

Boundary conditions with parameters

392

(1)

Green's tensors for systems of differential equations

393

(2)

Analytic continuation of the solutions of the equation Δ u + λu = 0

395

(1)

A theorem on the nodal curves of the solutions of Δu + λu = 0

395

(1)

An example of eigenvalues of infinite multiplicity

395

(1)

Limits for the validity of the expansion theorems

395

(2)

References

396

(1)

VI. Application of the Calculus of Variations to Eigenvalue Problems

397

(69)

Extremum properties of eigenvalues

398

(9)

Classical extremum properties

398

(4)

Generalizations

402

(3)

Eigenvalue problems for regions with separate components

405

(1)

The maximum-minimum property of eigenvalues

405

(2)

General consequences of the extremum properties of the eigenvalues

407

(17)

General Theorems

407

(5)

Infinite growth of the eigenvalues

412

(2)

Asymptotic behavior of the eigenvalues in the Sturm-Liouville problem

414

(1)

Singular differential equations

415

(1)

Further remarks concerning the growth of eigenvalues. Occurrence of negative eigenvalues

416

(2)

Continuity of eigenvalues

418

(6)

Completeness and expansion theorems

424

(5)

Completeness of the eigenfunctions

424

(2)

The expansion theorem

426

(1)

Generalization of the expansion theorem

427

(2)

Asymptotic distribution of eigenvalues

429

(16)

The equation Δu + λu = 0 for a rectangle

429

(2)

The equation Δu + λu = 0 for domains consisting of a finite number of squares or cubes

431

(3)

Extension to the general differential equation L[u] + λpu = 0

434

(2)

Asymptotic distribution of eigenvalues for an arbitrary domain

436

(7)

Sharper form of the laws of asymptotic distribution of eigenvalues for the differential equation Δu + λu = 0

443

(2)

Eigenvalue problems of the Schrodinger type

445

(6)

Nodes of eigenfunctions

451

(4)

Supplementary remarks and problems

455

(11)

Minimizing properties of eigenvalues. Derivation from completeness

455

(3)

Characterization of the first eigenfunction by absence of nodes

458

(1)

Further minimizing properties of eigenvalues

459

(1)

Asymptotic distribution of eigenvalues

460

(1)

Parameter eigenvalue problems

460

(1)

Boundary conditions containing parameters

461

(1)

Eigenvalue problems for closed surfaces

461

(1)

Estimates of eigenvalues when singular points occur

461

(2)

Minimum theorems for the membrane and plate

463

(1)

Minimum problems for variable mass distribution

463

(1)

Nodal points for the Sturm-Liouville problem. Maximum-minimum principle

463

(3)

References

464

(2)

VII. Special Functions Defined by Eigenvalue Problems

466

(80)

Preliminary discussion of linear second order differential equations

466

(1)

Bessel functions

467

(34)

Application of the integral transformation

468

(1)

Hankel functions

469

(2)

Bessel and Neumann functions

471

(3)

Integral representations of Bessel functions

474

(2)

Another integral representation of the Hankel and Bessel functions

476

(6)

Power series expansion of Bessel functions

482

(3)

Relations between Bessel functions

485

(7)

Zeros of Bessel functions

492

(4)

Neumann functions

496

(5)

Legendre functions

501

(5)

Schlafi's integral

501

(2)

Integral representations of Laplace

503

(1)

Legendre functions of the second kind

504

(1)

Associated Legendre functions. (Legendre functions of higher order.)

505

(1)

Application of the method of integral transformation to Legendre, Tchebycheff, Hermite, and Laguerre equations

506

(4)

Legendre functions

506

(1)

Tehebycheff functions

507

(1)

Hermite functions

508

(1)

Laguerre functions

509

(1)

Laplace spherical harmonics

510

(12)

Determination of 2n + 1 spherical harmonics of n-th order

511

(1)

Completeness of the system of functions

512

(1)

Expansion theorem

513

(1)

The Poisson integral

513

(1)

The Maxwell-Sylvester representation of spherical harmonics

514

(8)

Asymptotic expansions

522

(13)

Stirling's formula

522

(2)

Asymptotic calculation of Hankel and Bessel functions for large values of the arguments

524

(2)

The Saddle point method

526

(1)

Application of the saddle point method to the calculation of Hankel and Bessel functions for large parameter and large argument

527

(5)

General remarks on the saddle point method

532

(1)

The Darboux method

532

(1)

Application of the Darboux method to the asymptotic expansion of Legendre polynomials

533

(2)

Appendix to Chapter VII. Transformation of Spherical Harmonics

535

(11)

Introduction and notation

535

(1)

Orthogonal transformations

536

(3)

A generating function for spherical harmonics

539

(3)

Transformation formula

542

(1)

Expressions in terms of angular coordinates

543

(3)

Additional bibliography

546

(5)

Index

551

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Methods of Mathematical Physics, Volume 1

Summary

Author Biography

Table of Contents

Supplemental Materials